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A  TREATISE 


ON 


Linear  Differeintial  Equations 


THOMAS   CRAIG,   Ph.D. 

ASSOCIATE    PROFESSOR    IN    THE    JOHNS    HOPKINS    UNIVERSITY 
ASSOCIATE   EDITOR   OF   THE  AMERICAN  JOURNAL  OF   MATHEMATICS 


Volume  I 

EQUATIONS   WITH    UNIFORM    COEFFICIENTS 


NEW   YORK 

JOHN     WILEY     &     SONS 

15    AsTOR    Place 

1889 


Copyright,  1889, 

BY 

THOMAS  CRAIG. 


DR0MMOND  &  Neu,  Ferris  Bros., 

Electrotijpers,  Printers, 

1  to  7  Hague  Street.  326  Pearl  Street, 
New  York.  New  York. 


PREFACE. 


The  theory  of  linear  differential  equations  may  almost  be  said 
to  find  its  origin  in  Fuchs's  two  memoirs  published  in  1866  and 
1868  in  volumes  66  and  68  of  Crelle's  Journal.  Previous  to  this  the 
only  class  of  linear  differential  equations  for  which  a  general  method 
of  integration  was  known  was  the  class  of  equations  with  constant 
coefificients,  including  of  course  Legendre's  well-known  equation 
which  is  immediately  transformable  into  one  with  constant  coef- 
ficients. After  the  appearance  of  Fuchs's  second  memoir  many 
mathematicians,  particularly  in  France  and  Germany,  including 
Fuchs  himself,  took  up  the  subject  which,  though  still  in  its  infancy, 
now  possesses  a  very  large  literature. 

This  literature,  however,  is  so  scattered  among  the  different 
mathematical  journals  and  publications  of  learned  societies  that  it  is 
■extremely  difficult  for  students  to  read  up  the  subject  properly. 

I  have  endeavored  in  the  present  treatise  to  give  a  by  no  means 
complete  but,  I  trust,  a  sufficient  account  of  the  theory  as  it  stands 
to-day,  to  meet  the  needs  of  students.  Full  references  to  original 
sources  are  given  in  every  case. 

Most  of  the  results  in  the  first  two  chapters,  which  deal  with  the 
general  properties  of  linear  differential  equations  and  with  equations 
having  constant  coefficients,  are  of  course  old,  but  the  presentation 
of  these  properties  is  comparatively  new  and  is  due  to  such  mathe- 
maticians as  Hermite,  Jordan,  Darboux,  and  others.  All  that  fol- 
lows these  two  chapters  is  quite  new  and  constitutes  the  essential 
part  of  the  modern  theory  of  linear  differential  equations. 

The  present  volume  deals  principally  with  Fuchs's  type  of  equa- 
tions, i.e.  equations  whose  integrals  are  all  regular;  a  sufficient 
account  has  been  given,  however,  of  the  researches  of  Frobenius  and 
Thom^  on  equations  whose  integrals  are  not  all  regular.  A  pretty 
full  account,  due  to  Jordan,  has  been  given  of  the  application  of  the 


IV  PREFACE. 

theory  of  substitutions  to  linear  differential  equations.  This  subject 
will,  however,  be  very  much  more  fully  dwelt  upon  in  Volume  II, 
where  I  intend  to  take  up  the  question  of  equations  having  algebraic 
integrals  and  also  to  give  an  account  of  Poincare's  splendid  investi- 
gations of  Fuchsian  groups  and  Fuchsian  functions.  The  theory 
of  the  invariants  of  linear  differential  equations  has  been  several 
times  touched  upon  in  the  present  volume,  and  some  of  the  simpler 
results  of  the  theory  have  been  employed  ;  but  its  extended  develop- 
ment is  necessarily  reserved  for  Volume  II,  as  is  also  the  develop- 
ment of  Forsyth's  associate  equations,  about  which  extremely  inter- 
esting subject  very  little  is  as  yet  known. 

The  equation  of  the  second  order  with  the  critical  points  o,  i,  oo 
has  on  account  of  its  great  importance  been  very  fully  treated.  In 
connection  with  this  subject  it  seemed  to  me  that  I  could  not 
possibly  do  better  than  to  reproduce,  which  has  been  done  in  Chap- 
ter VII,  Goursat's  Thesis  on  equations  of  the  second  order  satisfied 
by  the  hypergeometric  series.  M.  Goursat  was  kind  enough  to  give 
me  permission  to  make  a  translation  of  his  Thesis,  which  is,  I  im- 
agine, not  very  well  known  among  English  and  American  students. 

In  Chapter  XIV  I  have  given  only  a  brief  account  of  equations 
with  doubly-periodic  coefficients.  I  intend,  however,  to  resume  this 
subject  in  Volume  II. 

I  wish  here  to  tender  my  thanks  to  M.  Goursat  for  his  kindness 
in  permitting  me  to  make  a  translation  of  his  most  valuable  Thesis, 
and  to  Dr.  Oskar  Bolza,  Mr.  C.  H.  Chapman,  and  Dr.  J.  C.  Fields 
for  much  valuable  assistance. 

T.  Craig. 

Johns  Hopkins  University, 
Baltimore,  1889. 


CONTENTS. 


CHAPTER   I. 

GENERAL   PROPERTIES   OF    LINEAR   DIFFERENTIAL   EQUATIONS. 


Two  General  Theorems, 
Normal  Form  of  a  System,     . 
Independence  of  Solutions,     . 
The  Adjoint  System, 
Application  to  a  Single  Equation, 
Lagrange's  Adjoint  Equation, 
Laguerre's  Invariants, 
Existence  of  an  Integral, 


PAGE: 

I 

3. 

4 

9 

II 

i8 

19 

22: 


CHAPTER   II. 

LINEAR   DIFFERENTIAL   EQUATIONS   WITH   CONSTANT   COEFFICIENTS. 

Euler's  Method,       .............  23; 

Hermile's  Presentation  of  Cauchy's  Method,  .......  24. 

Darboux's  Presentation  of  Cauchy's  Method,  .......  33 

Systems  of  Equations  with  Constant  Coefficients,  .......  36 

Equations  Reducible  to  Equations  with  Constant  Coefficients,        ....  41 

Meray's  Method, 43 

CHAPTER   III. 

THE   INTEGRALS    OF   THE    DIFFERENTIAL   EQUATION 


dx»-       ^  dx  "  -  I 


■J". 


d" 


dx' 


+ 


+  /«7  =  o- 


Linear  Transformation  of  a  System  of  Fundamental  Integrals,       ....  51 

Equations  with  Doubly-Periodic  Coefficients,  •■....»,  52 

Values  of  the  Cofficients />  1,  p  2,  .  .  .  ',  pn  ,        .        ■         .         .         .         .         .         .  54 

Formation  of  a  System  of  Fundamental  Integrals,  ......  56 

The  Characteristic  Equation,  •••.......  59 

Independence  of  Choice  of  Fundamental  System,   .......  61 

Hamburger's  Theorem,  ...........  63 

Case  of  Equal  Roots,       ••..•.......  64. 

V 


■VI  CONTENTS. 

PAGE 

Investigation  of  the  Forms  of  the  Integrals,   ........  65 

Hamburger's  Determination  of  the  Sub-groups  of  the  Integrals,     .         .         .         .76 

Jordan's    Canonical    Form   of   the    Substitution    corresponding    to   any    Critical 

Point, 84 

Forms  of  the  Substitutions  S  and  S'  in  the  case  of  Equations  with  Doubly-Periodic 

Coefficients,     .............  84 

Regular  Integrals,                     91 

CHAPTER   IV. 

FROBENIUS'S   METHOD. 

Convergence  of  a  Series  and  Proof  of  the  Existence  of  an  Integral,         ...       94 
Existence  of  Logarithms 104 

CHAPTER   V. 

LINEAR   DIFFERENTIAL   EQUATIONS    ALL   OF    WHOSE    INTEGRALS    ARE    REGULAR. 

T)efinition  of  the  Function  F, 108 

Properties  of  Regular  Functions,  .........     109 

Fuchs's  Theorem,  .         .         .         •         .         .         .         •         .         .         .         .111 

The  Indicial  Equation,    .  .  •  .  .  .  .  .  .  .  .  .118 

Forms  of  the  Coefficients 123 

Fuchs's  Converse  Theorem  and  Existence  of  an  Integral 123 

Sum  of  the  Roots  of  all  the  Indicial  Equations 134 

Applications, I35 

Reducibility  and  Irreducibility 148 

CHAPTER   VI. 

LINEAR   DIFFERENTIAL   EQUATIONS   OF   THE   SECOND   ORDER,    PARTICULARLY   THOSE    WITH 

THREE   CRITICAL   POINTS. 

Fuchs's  Transformation,         .         .         .         .         .         .         .         .         •         •         -154 

The  General  Linear  Transformation, 156 

The  Critical  Points  o,  I,  CO I57 

Change  of  the  Dependent  Variable 158 

Differential  Equation  for  the  Hypergeometric  Series I59 

The  Twenty-four  Particular  Integrals, iC)4 

Division  of  these  into  Six  Groups, 166 

Relations  between  the  Integrals  belonging  to  two  different  Critical  Points,    .         .169 

Markoff's  Two  Problems I74 

Heun's  Application  of  Abelian  Integrals  of  the  Third  Kind I77 

Riemann's  P-function,   .......•••••      185 

Spherical  Harmonics,  Toroidal  Functions,  and  Bessel's  Functions,  .         .         .193 

Generalized  Spherical  Harmonics  and  Bessel's  Functions, 200 

Humbert's  Investigation, 202 


CONTENTS.  VU 
CHAPTER   VII. 

ON  THE    LINEAR    DIFFERENTIAL    EQUATION    WHICH    ADMITS    THE   HYPERGEOMETRIC 
SERIES   AS   AN   INTEGRAL  ;    BY  M.   EDOUARD  GOURSAT. 

Part  First. 

PAGE 

Summary  of  Contents,    ............  212 

Jacobi's  Method, 215 

Table  of  Integrals, 229 

Application  of  Cauchy's  Theorem  and  Relations  between  the  Integrals,         .         .  232 

Application  to  the  Complete  Elliptic  Integral  of  the  First  Kind,     ....  252 

Schwarz's  Results,           ............  258 

Part  Second. 

Transformations  of  the  Hypergeometric  Series, 276 

Tannery's  Theorem,        ............  277 

Change  of  the  Independent  Variable:  Linear  Transformation,         ....  279 

General  Theory  of  the  Direct  and  Inverse  Transformations,  .....  281 

CHAPTER   VIII. 

IRREDUCIBLE    LINEAR    DIFFERENTIAL   EQUATIONS. 

Frobenius's  Theorems,             ...........  362 

Determination  of  the   Reducibility  or  Irreducibility  of  a  Linear  Differential  Equa- 
tion by  the  Study  of  its  Group,         .........  369 

CHAPTER  IX. 

LINEAR    DIFFERENTIAL    EQUATIONS    SOME   OF   WHOSE   INTEGRALS   ARE   REGULAR. 

Definition  of  the  Order  of  a  Coefficient  of  the  Equation,         .....  373 

The  Characteristic  Index,        ...........  374 

The  Indicial  Equation,   ............  374 

Thome's  Theorems,         ............  375 

The  Characteristic  Function,            ..........  377 

The  Indicial  Function,    ............  379 

Superior  Limit  to  the  Number  of  Linearly  Independent  Integrals,         .         .         .  382 

The  Normal  Form  of  a  Linear  Differential  Equation,     ......  382 

Composite  Differential  Quantics,     ..........  383 

Irregular  Integrals,           ............  386 

Thome's  Normal  Integrals,     ...........  388 

CHAPTER   X. 

DECOMPOSITION   OF   A   LINEAR    DIFFERENTIAL    EQUATION   INTO  SYMBOLIC    PRIME  FACTORS. 

Definition  of  Symbolic  Prime  Factors,   .........  389 

Method  of  Decomposition,     ...........  39a 

Resulting  Forms  of  the  Coefficients  of  the  given  Equation,     .....  392 

A  Transformation  of  the  given  Equation,         ........  393 


Vlll  CONTENTS. 

PAGE 

Conjugate  Solutions, 398 

Conditions  for  Commutative  Prime  Factors,            .......  400 

Form  of  Linear  Differential  Equation  possessing  Commutative  Prime  Factors,       .  402 

Application  to  Regular  Integrals, 406 

Composite  Differential  Quantics  in  general,    ........  409 

Number  of  Linearly  Independent  Regular  Integrals, 415 

Theorems  concerning  the  Adjoint  Equation,  .         .         .         .         .         .         .         .421 

CHAPTER   XI. 

APPLICATION   OF   THE   THEORY    OF   SUBSTITUTIONS    TO  LINEAR  DIFFERENTIAL    EQUATIONS. 

Products  and  Powers  of  Substitutions 422 

Canonical  Form,      .............  423 

Function-Groups,             ............  424 

Number  of  Distinct  Functions  in  a  Function-Group, 425 

Prime  Groups,                  ............  432 

Characteristic  Equation  possessing  a  single  Root, 433 

Case  of  Incompatible  Equations,     ...........  444 

Canonical  Form  of  Substitutions  in  a  Non-Prime  Group 446 

Final  Determination  of  Function  Groups, 448 

CHAPTER   XII. 

EQUATIONS   WHOSE    GENERAL    INTEGRALS   ARE    RATIONAL. — HALPHEN's    EQUATIONS. 

Equations  whose  General  Integrals  are  Rational 452 

Form  of  the  General  Integral, 453 

Halp hen's  Equations. 
(i)  Equations  whose    Integrals  are  Regular   and   Uniform  in  every  region  of  the 

plane  that  does  not  contain  the  point  x^, 454 

(2)  Equations  whose  General  Integral  is  of  the  form 

y  =  f:^«.-V"i(-*")  +  ^^^^-^M^)  4-  •   •  •  +^»^"»V«W 455 

Examples,       . 4^1 

CHAPTER   XIII. 

TRANSFORMATION  OF  A  LINEAR  DIFFERENTIAL    EQUATION. — FORSYTH'S    CANONICAL  FORM. 

— ASSOCIATE    EQUATIONS. 

Change  of  the  Dependent  Variable  and  Removal  of  the  Second  Term,            .         .  463 

Change  of  both  Variables, 4^4 

Brioschi's  Invariant,        ............  4^7 

Invariants  of  a  Linear  Differential  Equation, 469 

Index  of  an  Invariant. 469 

Forsyth's  Canonical  Form, 470 

Lagrange's  Adjoint  Equation 47i 

Forsyth's  Associate  Equations, 473 


CONTENTS. 


IX 


Properties  of  Lagrange's  Adjoint  Equation,    . 
Adjoint  Differential  Quantics,  .... 

The  Adjoint  of  a  Composite  Differential  Quantic, 
Relation  between  the  Integrals  of  Adjoint  Equations, 
Frobenius's  Associate  Bilinear  Differential  Quantic, 

Halphen  on  Adjoint  Quantics 

Self-Adjoint  Quantics  and  Equations, 

Examples,        ........ 

Appell's  Theorem,  ...... 


PAGE 

479 

482 
484 

485 
487 
488 
490 
490 
495 


CHAPTER   XIV. 

LINEAR   DIFFERENTIAL    EQUATIONS    WITH    UNIFORM    DOUBLY-PERIODIC    COEFFICIENTS. 

Elementary  Properties  of  Doubly-Periodic  Functions,    ......  496 

Integrals  of  Equations  with  Doubly-Periodic  Coefficients,       .  .         .         .         .  503 

Picard's  Theorem,  ............  503 

Canonical  Form  of  the  Substitutions  6"  and  y 504 

Formation  of  Functions  submitting  to  these  Substitutions  and  satisfying  the  given 

Differential  Equation,        ...........  504 

Determination  of  the  Constants  entering  into  the  Integrals,    .....  510 

Determination  of  Additional  Integrals,  ........  514 


LINEAR    DIFFERENTIAL   EQUATIONS. 


CHAPTER    I. 

GENERAL   PROPERTIES   OF   LINEAR   DIFFERENTIAL   EQUATIONS. 

According  to  general  usage  we  will  denote  by  x  an  independent 
complex  variable ;  that  is,  x  is  the  affix  of  any  variable  point  in  the 
plane.  When  fixed  points,  such  as  the  critical  points  of  the  various 
functions  we  shall  encounter,  are  to  be  spoken  of,  we  will  generally 
denote  their  affixes  b)^  a,  b,  c,  .  .  .  Again,  j  is  an  unknown  function 
of  X,  defined  by  a  linear  differential  equation — that  is,  a  differential 
equation  whose  left-hand  member  is  a  linear  function  of  y  and  of 
the  derivatives  of  y  with  respect  to  x ;  the  coefficients  in  this  linear 
function  are  arbitrary  functions  of  x;  the  right-hand  member  of  the 
linear  differential  equation  is  either  zero  or  a  function  of  x  alone. 
The  equation  has  thus  the  form 

A  first  general  property  of  linear  differential  equations  is  that 
they  remain  linear  when  we  change  the  independent  variable.  Write 
viz.,  X  —  (pit),  and  form  the  new  differential  equation  connecting  jj/ 
and  t:  it  will  obviously  be  an  equation  of  the  same  form  as  (i), 
having  new  coefficients  containing  t  alone.  The  verification  of  this 
is  so  simple  that  it  need  not  be  given  here. 

Another  general  proposition  is  (see  Jordan's  Conrs  d Analyse, 
vol.  iii.  p.  1 36)  as  follows :  If  y,  2,  .  .  .,  w  denote  n  functions  of  tht 
same  independent  variable  x,  ajid  satisfying  the  system  of  n  lineaf 
differential  equations, 

(2)  ^,  =  o,         E^^o,  .  .   .,  E„  =  o\ 


2  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  if  V  is  a  polynomial  in  y,z,  .  .  .,  w  and  their  successive  derivatives, 
the  different  terms  of  which  have  for  coefficients  arbitrary  functions 
of  x:  then  V  will  satisfy  a  linear  differential  equation  whose  coefficients 
are  rationally  expressible  in  terms  of  the  coefficients  of  E^,  .  .  .  E,„ 
V,  and  of  their  successive  derivatives. 

Let  pi,  fx' ,  .  .  .  denote  the  highest  orders  of  the  derivatives  oi  y, 
z,  .  .  .  ,  respectively,  in  equations  (2),  and  let  A  denote  the  degree  of 
the  polynomial  V.  Form  now  the  successive  derivatives  of  V  with 
respect  to  x\  each  of  these  is  a  polynomial  of  degree  A  \\\  y,  z,  .  .  . 
and  their  derivatives ;  and,  further,  the  derivatives 

-  di^y        di^-^^y  d*^' z       d'^'  +  '^z 

^^^  ^^ '      dx^~^  ^      ■  '  ,      -^o       -^V+l  '•••'••• 

are  linearly  expressible  in  terms  of  the  derivatives  of  lower  order  by 
means  of  equations  (2)  and  their  derivatives.  Substituting  these 
values  for  the  quantities  (3),  we  have 


(4) 


dx 


where  the  functions  X,  X' ,  .  .  .  are  functions  of  x  of  the  kind  men- 
tioned in  the  above  proposition,  and  P^,  P^  ,  .  .  .  are  products  of 
the  form 

dy\-^  Id^-^  yYy.-^     Jdz^^  fd'''-' sVt^'-i 


^\dxl    '  •  •  •  '   \dx>^-^}         '      \dx)   '   •  •  •  '    \dx^' 

the  total  number  of  factors  (taking  account  of  their  orders  of 
multiplicity)  being  at  most  =  A.  The  number  of  these  products  P 
is  of  course  limited  ;  let  them  be  denoted  by 

-*  1  )      P-ii  '  '  -1  Pi- 

Eliminating  these   quantities    between   equations  (4),  we  obviously 

arrive  at  a  Imear  relation  connecting    V,  --— ,  --^- ,  .  .  .     rrom  the 

dx     dx 

theory  of  ordinary  differential  equations  we  know  that  any  simul- 
taneous system  of  such  equations  can,  by  the  introduction  of  certain 


GENERAL   PROPERTIES.  3 

auxiliary  functions  and  the  performance  of  certain  simple  algebraical 
operations,  be  thrown  into  the  normal  form — that  is,  can  be  replaced 
by  a  system  of  equations  in  each  of  which  there  appears  only  the 
first  derivative  of  one  of  the  (old  and  new)  dependent  variables.  In 
case  the  given  system  of  simultaneous  equations  is  linear,  it  is  ob- 
vious that  the  property  of  linearity  will  be  retained  when  the  equa- 
tions are  replaced  by  the  normal  system.  In  studying,  then,  a 
system  of  simultaneous  linear  differential  equations,  we  may  at  once 
suppose  the  system  to  be  replaced  by  the  corresponding  normal 
system.     Suppose  such  a  normal  system  to  be 


<5) 


dx 

dx 

df^ 
dx 


1     ^7iny n  —    -^  n  f 


where  T^  ,  T^ ,  .  .  .  ,  T„  and  the  coefficients  atk  are  functions  of  x. 
As  a  particular  case,  suppose  the  given  system  of  simultaneous 
linear  differential  equations  to  reduce  to  the  single  one, 

d"v  d"~'^v  d"~'^v 


Denoting  by  y',  y" ,  .  .  .  ,  jv"~'  new  auxiliary  dependent  variables, 
we  can  replace  (6)  by  the  equivalent  normal  system, 


<7) 


r  ^y 

dx 

dy 

dx 
dy" 


-y 
-y' 


=  o, 
=  o, 


dx 
dy"-'' 
^    dx 


-  y 


_J^Pjn-.J^ 


=  o, 

+AJ  =  T. 


4  LINEAR   DIFFERENTIAL   EQUATIONS. 

As  the  reader  is  supposed  to  be  familiar  with  the  elementary 
theory  of  differential  equations,  and  with  the  general  properties  of 
linear  differential  equations,  it  will  be  sufficient  here  to  state  some 
of  these  properties  without  proof.  First:  ifjAjJ's',  .  .  .  ,  yn  ',  y^i  y^y 
.  .  .  ,  yn  ;  •  •  •  are  particular  solutions  of  equations  (5)  when  T^ , 
T^,  .  .  .  ,  T„  are  all  supposed  to  be  zero,  then,  denoting  by  C^  ^ 
C^,  ,  .  .  ,  C„  arbitrary  constants, 


(8) 


C.y:  +  C,y:  +  .  .  .  +  C,.y„\ 
C,y"  +  C,y,'  +  .  .  .  +  C„yu, 


are  also  solutions.  This  is  at  once  proved  by  substituting  in  equa- 
tions (5)  when  the  right-hand  members  are  all  replaced  by  zero,  that 
is,  when  equations  (5)  are  of  the  form 


(9) 


dx 

dx 

dy^ 
dx 


+  ^.1  J.   +  «.2A  +    •    •    •    +  «2«J«    =   O, 


+  «/nJ,  +  ««27.  +    •    •    •    +  ^»nyn 


O. 


An  obvious,  but  useless,  solution  of  these  last  equations  is  j,  =  y^:= 
.  .  .  =  jj/„  =  o.  Aside  from  this  useless  solution,  suppose  equations 
(9)  to  admit  the  following  k  particular  solutions : 


(10) 


y:, 
y:, 


y:, 


y.^ 


yn 
yn 


y 


I    k 


These  solutions  will  be  said  to  be  independent  if  one  at  least  of  the 
determinants  of  order  k  formed  from  the  different  columns  of  (10) 
does  not  vanish.  If  /'  =  n,  then  the  solutions  (10)  will  be  independ- 
ent if  the  determinant 


GENERAL   PROPERTIES. 


(II) 


y: 


y.> 
y:, 


y",     y.", 


yn, 

yn, 

yn", 


does  not  vanish.     The  particular  application  of  this  to  equations  (7) 
and  (6)  is  easily  seen,  but  will  be  referred  to  later  on. 

As  an  illustration  of  another  well-known  general  theorem,  stated 
below,  consider  the  system 


(12) 


dy 
dx 

dz 

^  +  «.;^  +  b,z  +  c,u  +  d,v 


-\-  ay  -\-  dz  -\-  cii  -\-  dv       =0, 


o, 


dii 
~dx 
dv 
dx 


-.^-  ^.y^  K^  +  ^=?^  +  <^'  =^  o, 


+  ^.y  +  ^3-  +  ^3?^  +  d,- 


o, 


and  suppose  that  we  know  two  independent  solutions  of  this  sys- 
tem, viz.: 

y,,     z^,     z/, ,     v^ ; 


y. 


,     i^o ,     V, 


Suppose,  for  example,  that  the  determinant 
does  not  vanish.     Let  us  write  now 


(13) 


y  =  C,y,  +  C^y^ , 

z  —  c  -  4-  r  - 

u  =  c,u,  +  c^u^  +  a, 


where  £^f,,  ^,  77,  are  new  functions  of  x.     Substituting  the  values    /  U  ^ 
(13)  in  (10),  and  remarking  that  the  terms  in  c^  and  <f,  vanish  (since 


6  LINEAR  DIFFERENTIAL  EQUATIONS. 

equations  (lo)  would  be  satisfied  if  B,  and  -q  were  zero  and  C^  and  C^ 
constants),  we  have 


(H) 


dC, 


dC„ 


dC,  dC,  c    ,     r 


=  o, 
=  o, 


'-dx  ''^  +  "di  ''-^  +  i^  +  ^^^  +  ^='^  =  ^' 


dC, 
dx 


dx 

dn 


dC^  dC^  --  , 

L^^^'  +  -dx"^  -^  dx  +  '^^+  <^  =  °- 


Solving  these  for  -r-^  ,    -7-^ , 


-y-  ,  we  derive  the  system 


dx  ^    dx  ^    dx''    dx 


^-1^=^^+^^' 


(15) 


dC^ 
dx 


=  A,B  +  ^.77, 


=     /^3^    +^3^. 


From  the  last  two  equations  we  may  suppose  5"  and  7;  to  be  obtained, 
and  then  from  the  first  two  equations  we  find  C^  and  C^  by  quadra- 
tures. Knowing  then  two  independent  solutions  of  the  system  (12) 
of  four  equations,  we  have  made  the  integration  of  this  system 
depend  on  the  integration  of  a  similar  system  of  two  linear  differ- 
ential equations,  viz.,  the  last  two  of  equations  (15),  and  on  quad- 
ratures. We  can  now  show  that  the  new  solutions,  say  B,^  and  if  y 
form,  with  the  given  solutions 


jl/,,     ^,,     11^,     v^, 

an  independent  system.  (We  of  course  exclude  the  obvious  solu- 
tion, ^  =  o,  7  =  o,  of  the  last  two  of  equations  (15)).  To  fix  the 
ideas,  suppose  ^'  to  be  different  from  zero,  and  let  C^  and  (7/  denote 


►    GENERAL   PROPERTIES.  7 

the  values  of  (T,  and  C^  as  obtained  by  the  quadratures ;  then  we  can 
show  at  once  that  the  solution 


(i6) 


-'3     '-'1   '^1        1^     *-'2  '^2  » 


is  independent  of  the  two  known  solutions, 


y^ 


in  fact,  we  see  that  the  determinant 


Jl 

'   "^i 

?^ 

y.,    -2 

,           ?/ 

y,.   -3 

,       7( 

?* 

J3» 

^1  > 

J2, 

•^2    ) 

reduces  to 


and  by  hypothesis  neither  of  these  factors  vanishes.  The  general 
theorem  above  referred  to,  and  of  which  the  preceding  is  a  simple 
illustration,  may  be  stated  as  follows  : 

If  zue  know  k  independent  solutions  {k  <  Ji)  of  a  system  of  linear 
differential  equations  of  t lie  first  order,  the  integration  of  this  system 
can  be  condiicted  to  that  of  an  analogous  system  of  n  —  k  equations 
and  to  quadratures.  Eaeh  solution  of  this  neiv  system  will  furnish 
a  solution  of  the  given  system  zvJiich  will  be  independent  of  those 
already  known. 

Another  well-known  general  theorem  which  may  be  stated  with- 
out proof  is:  Every  system  of  linear  differential  equations  of  the 
form  (9)  admits  n  particular  independent  solutions, 


Ji>    J2 


yn^ 

yn, 


Ji"»  y-:', '  •  • ,   yn" 


8  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  its  most  general  solution  is 

c,y"  +  c,y^  +  .  .  .  +  c„y», 

(^.y.'  +  c.y'  +  •  •  •  +  ^«j.% 

^^xJn  +  c^y„^  +  .  .  .  +  c„j„«  ; 

where  r, ,  c^,  .  .  . ,  c,i  arc  arbitrary  constants. 

This  is  readily  verified  by  substitution  in  (9),  and  by  aid  of  the 
preceding  theorem. 

Suppose 


(17) 


r  y:  =c:y:  +c:y:  + 
v:  =c:y:  +  c:y:  + 

Yn  ^  c,'y,"  +  c^y,"  + 

Yn    =  c^'y:  +  c^^y:  + 
Vn    =  c:'y:  +  c^-y:  + 

Y„"  =  c,"y,''  +  f,«j)//  + 


+  <^n    yn\ 

+  c„'y„% 
+  c„"y„\ 


to  be  n  arbitrary  particular  solutions  of  equations  (9).     The  deter- 
minant 

I  F,*  I      {i,  k=i,2,...n) 

is  obviously  equal  to  the  product  of  the  determinants 

I  j/  I       and       1  Ci^  I      (/,  k  =  1,2,  .  .  .  n). 

Since  the  determinant    |  yi^  \    is  by  hypothesis  not  zero,  it  follows 
that  in  order  that  the  determinant 

I  Y,''  I 


may  be  different   from   zero,  and  consequently  that  the  solutions 


GENERAL   PROPERTIES.  9 

(17)  may  form  an  independent  system,  we  must  have  the  constants 
d''  so  chosen  that  the  determinant 

shall  not  vanish. 

To  every  system  of  linear  differential  equations  whose  right-hand 
member  is  zero,  such  as 


<i8) 


dx 

^  +  «.i Ji  +  ^.2^2  +  .  .  .  +  a,„y„ 


+  ^nJi   +  ^nJ.  +  •    •    •    +  ^i«J«     =   O, 


o, 


+  ^«»  J/i  =  o. 


there  is  associated  a  system  defined  by  the  equations 


'(19) 


dV„ 


-  +  «n  ^1   +  ^..  5^.  +   •    •     •   +  ^«i  ^n 


+  ^u  ^\  +  ^«  ^.  +  •  •  •  +  ^«2  y„ 


o, 
o, 


dV„ 
dx 


+  a,„  F,  4-  ^3„  K,  +  .  .  .  +  a„„  V„  =  o. 


This  system  (19)  will  be  called  the  adjunct  system  to  (18).  A 
very  beautiful  relation  exists  between  the  solutions  of  equations 
(18)  and  (19).  Suppose  equations  (18)  to  be  multiplied  by  Fj , 
K; ,  .  .  . ,  F„,  respectively,  and  equations  (19)  to  be  similarly  multi- 
phed  by  —  jj/j,  —  jj/,,  .  .  .,—/„;  add  together  the  results  of  these 
two  multiplications  and  we  find,  after  making  certain  simple  reduc- 
tions, 


F^.    F^4-  4-  F 


dyn 
dx 


.    4,,    dv,  dv„ 


y 


lO 


LINEAR  DIFFERENTIAL   EQUATIONS. 


or 

^UJ".+  J.F,  +  .  .  .  +  ynY„)  =o; 
and  consequently 
(20)  J.  5^,  +  ^.  ^«  +  •  •  •  +  J«  5^«  =  const. 

It  is  now  obvious  that  the  complete  solution  of  either  of  the 
systems  (18)  or  (19)  will  involve  the  complete  solution  of  the  other 
system.     Suppose,  in  fact,  that  we  know  n  independent  solutions, 

fr\  y.\  •  ■  •  ,  yn, 
jiS  y.\  •  •  •  .  yn, 


of  equations  (18);  then  denoting  by 


yn* 


^1»    ^2 


n  arbitrary  constants,  we  have 


>    ^n  J 


r  y.y:  +  y.y:  + 


+     Y,J'n     =   C^  , 


(21) 


!  y.y:+  y.y:+  •  •  •  +  yny:  =  c^, 


Y,y--\-  y,y,"  + 


+     Ynyn"  =   C„. 


Solving  these  last  equations  for  Fj ,  F, ,  .  .  .  ,  Y„ ,  we  have  the 
general  solution  of  the  system  (19)  involving,  as  it  should,  n  arbitrary 
constants.  If  we  knew  only  k  solutions  {k  <  n)  of  the  given  system 
(18),  then  we  would  have  the  ^  relations 


(22) 


(y,y:+  y.y:+  •  •  •  +  y.yn  =c,, 
y.y:+  y.y:+  •  •  •  +  y^yj^c,, 


[y.y.'+  Ky.'  + 


+  F„j//  =  c^. 


GENERAL   PROPERTIES.  II 

These  last  relations  would  enable  us  to  eliminate  k  of  the  func- 
tions Fj ,  y^j  •  •  •  >  F«  from  the  adjunct  system  (19),  and  so  the 
problem  of  the  integration  of  this  system  would  be  reduced  to  that 
of  the  integration  of  a  similar  system  containing  only  n  —  k  equa- 
tions. This  matter  of  the  adjunct  equation  will  be  taken  up  again 
in  connection  with  the  study  of  equation  (6)  and  the  corresponding 
system  (7),  when  we  will  assume  for  these  equations  that  T,  the 
right-hand  member  of  (6),  is  equal  to  zero. 

As  we  shall  have  but  little  to  do  with  linear  differential  equa- 
tions whose  second  members  are  different  from  zero,  and  as  the 
general  properties  of  such  equations  are  sufficiently  well  known,  we 
need  only  state  here  two  fundamental  propositions  concerning  them. 

Supposing  the  linear  differential  equation  whose  right-hand  mem- 
ber is  other  than  zero  to  be  of  the  form 

d"y  d"-^  d"-^y 

<^^^  dx-  +^"^^^^  +^V.r^^  +  .  .  .  +  A7  =  H, 

where  H  is  a  function  of  x\  then  if  Fbe  a  particular  solution  of  this, 
equation,  and 

^1  J.  +  ^2  J.  +  •  •  .  +  c„yn 

the  general  solution  of  the  equation  when  H  is  replaced  by  zero,  we 
have  as  the  complete  solution  of  the  given  equation 

1^  =  ^.Ji  +  ^.72  +  •  •  •  +  c,,y„  -f  F. 

Again,  the  integration  of  the  equation  {a)  of  order  n  can  be  made 
to  depend  upon  the  integration  of  a  similar  equation  of  order  n  —  k, 
provided  we  know  k  particular  integrals  of  {a).  Proofs  of  these 
theorems  will  be  found  in  Baltzer's  Theoric  dcr  Deterinmaiiten, 
Houel's  Coiirs  d' Analyse,  Forsyth's  Differential  Equations,  etc. 

We  will  now  go  on  to  the  study  of  the  linear  differential  equa- 
tion 

,  d"y  d*''  'jj/  d"-''y 

^'3^  ^«  +  ^^d^-^  +  ^^d^-^  +  .  •  .  +  Pny  =  o. 


12 


LINEAR   DIFFERENTIAL   EQUATIONS. 


We  have  seen,  equation  (7),  that  by  introducing  new  auxiliary- 
variables  n  —  I  in  number,  say 

■n',     ?/',  .  .  .  ,  if-\ 

we  can  replace  this  last  equation  by  the  system 


(24) 


dx 
d,/ 
dx 


V 


—  V 


=  O, 
=  o. 


dt/" 


dx 

dff  ~ 
dx 


—  r/" 


O, 


'  +  Av"-'  +  A'/"-^  +  .  .  •  +  A7  =  o- 


Let 


(25) 


y.,  //,  j/ 


7/' 


denote  71  independent  solutions  of  this  system  ;  then  the  determinant 


(26) 


}'n  ,        yn 


y/' 


yn' 


must  not  vanish.     It  is  perfectly  obvious  from  the  form  of  equations 
(24)  that 


y.  = 


dy, 
dx' 


n_<£y^ 

^'     ~   dx'  ' 


y^< 


d"  -  y„ 
dx"  -  ' 


that  is,  all  the  integrals  y  are  derivatives  of  the  corresponding  inte- 
grals f,  and  also  that  y^,  y^,  •  .  •  ,  /«  are  independent  integrals  of 
'(23) ;  the  condition  then,  by  hypothesis,  that  a  system  of  integrals  of 


(23),  viz.x,  J,, 
minant 


GENERAL   PROPERTIES.  1 3. 

,  yn ,  shall  be  independent,  is  that  the  deter- 


(27) 


d"  -  y,     d*"  -  y, 

dx""  - ~  '    dx"^^'    •  •  •  . 


^«  -  ^j|/„     ^«  -  yn 


dx" 


dx" 


y. 


7. 


yn: 


,  =  A 


must  not  be  =  o. 

The  nature  of  the  independence  of  these  integrals  must  now  be 
shown.  The  non-vanishing  of  D  does  not  mean  that  the  functions 
Ji  >  J2 '  •  •  •  ^  yn  are  absolutely  independent,  but  only  that  they  are 
linearly  independent ;  that  is,  if  D  is  not  zero,  then  there  cannot  exist 
a7iy  linear  relation  of  the  form 


(28) 


^1/1  +  ^.J.  + 


+  c„y„  =  o ; 


where  c^,  c^,  .  .  .  ,  c„  are  arbitrary  constants.  On  the  other  hand^ 
\i  D  =■  o,  then  some  relation  of  the  form  (28)  exists  in  which  the 
constants  c^,  c^,  .  .  .  ,  c„  are  not  all  zero. 

The  following  proof  of  these  theorems  is  due  to  Frobenius,  and 
is  entirely  independent  of  any  considerations  involving  the  linear 
differential  equation  : 

Suppose  y^,  y^,  •  •  .  ,  y„  to  he  defined  as  series  going  according 
to  positive  integral  powers  of  x  —  a,  and  all  convergent  inside  a 
circle  of  radius  p,  and  having  its  centre  at  the  point  x  =  a.  The 
variable  x  is  now  of  course  restricted  to  the  interior  of  this  circle. 
If  among-  these  functions  there  is  a  linear  relation  of  the  form 


(29) 


^.yi  +  ^.y^  + 


+  ^«  j«  =  o, 


where  c^,  c^,  .  .  .  ,  c„  are  constants,  then  between  this  equation  and 
its  ;z  —  I  derivatives  v/e  can  of  course  eliminate  the  constants  c,  and 
as  a  result  have 


H 


LINEAR  DIFFERENTIAL   EQUATIONS. 

d"  -  y^      d"  -  y^ 


(30) 


z>  = 


dx"  -  ' '     dx"  -'  '    •  *  ■  ' 

d"  -  y^     d"  -  y^ 

dx"  -  ' '     dx"  -  '  »    •  •  •  » 


d^  -y^     d^_y» 
dx"  -  '  '    dx"  -  ^  ' 


Ji. 


J. 


yn, 


=  o. 


d^v  • 
For  brevity  we  will  write,  as  usual,  jj'^  instead  of  -7-^-     ^^^  order 

to  prove  the  converse  of  the  theorem,  that  is,  to  prove  that  if  Z>  =  o 
there  exists  a  linear  relation  of  the  form  (29)  between  the  functions 
J^ij  J2)  •  •  •  .  J«)  we  will  first  assume  that  the  minor  D,i"  ~  \  corre- 
sponding to  the  element  y„"  ~  '  of  D,  is  not  identically  zero.  Con- 
sider now  the  system  of  ;^  —  i  linear  equations 


{31) 


-■J^i         +  -2J.         +  .  .  .  +  s„y„         =  o, 
.-.f"  -  '  +  s,y,"  --+...+  s„y,;'  -'  =  0. 


Solving  these  for  the  unknown  quantities  ^1,  s^,    .  .  .  ,  2„,  we  have 
for  the  ratios 


perfectly   determinate    finite   functions ;    but,    since   by   hypothesis 
D  =  O,  these  functions  must  satisfy  the  equation 


(32) 


^1^/ 


+  -.J."  -'+...+  ^„ j«''  -  ^  =  o. 


Taking  this  equation   into  account,  we  have,  as  is  easily  seen,  by  |] 
differentiating  equations  (31),         '    "    , 


(33) 


+  ^.y:    +  . . .  +  ^M    =  o, 


c-  '-1/  «  -  2  _4_  r  ' V  «  -  2  -L  — U  p.   ' v  n  —  1  —  (-)« 


GENERAL  PROPERTIES.  1 5 

dz 
in  which  z'  denotes  -^.     Now  since  equations  (31)  determine  defi- 

initely  the  ratios 

z^        z,  z"  - '' 

1~ '     1~  >     •  •  •  >     ~z      > 


equations  (33),  which  are  identical  in  form  with  equations  (31),  must 
equally  give  perfectly  determinate  finite  values  for 


and  these  values  are  respectively  equal  to  the  values  of 


that  is,  we  have 

Zji        Zk  d    ^k 

—  =  — />    or    T^;:  T-  =  o  (^  =  I'  2,  .  .  .  ,  «  -  i), 


or 


■'n  ^n 


jwhere  c^,  c^,  .  .  .  ,  c„  are  constants  of  which  c„  is  arbitrary  but 
different  from  zero.     From  the  first  of  equations  (31)  we  have  now 

1(34)  c,y,  +  c,y^  4-  .  ,  .  -|-  c„y„  =  o. 

Let  us  now  suppose  I),"~^  =  o,  but  the  minor  of  order  n  —  2, 
^,"_\'  of  DJ*  ~  ' ,  which  corresponds  to  the  element  j/_~"/,  to  be  dif- 
ferent from  zero  ;  then  by  proceeding  in  the  same  manner  we  arrive 
at  a  relation  of  the  form 

(35)  ^iJi  +  ^.7.  +  .  •  •  +^«-.J«-i  =  o; 

in  which  c„_^  is  not  zero.  Continuing  in  this  way,  we  can  show  that 
if  Z)  =  o  there  is  always  a  relation  of  the  form  (34),  in  which  the 
constants  c^,  c^,  .  .  .  ,c„  are  not  all  zero ;  remembering,  of  course,  that 
the  equation  jj  =  o  is  to  be  included  in  this  form. 


1 6  LINEAR   DIFFERENTIAL  EQUATIONS. 

We  will  speak  in  future  of  any  number  of  functions  being  inde- 
pendent if  there  exists  between  them  no  linear  homogeneous  relation 
with  constant  coefficients,  and  the  condition  for  this  independence  is 
expressed  in  the  theorem  : 

If  n  functions  y^,  y^,  •  •  .  ,  jF«  cire  independent,  tJicn  their  de- 
terminant {i.e.,  the  determinant  D)  does  not  vanisJi.  Conversely,  if 
the  funetions  are  not  independent,  their  determinant  vanisJics. 

The  theorem  has  only  been  proved  for  the  region  of  the  plane 
inside  the  circle  of  radius  p  and  centre  x  ^=  a,  but  by  a  well-known 
theorem  in  the  theory  of  functions  the  truth  of  the  theorem  is  estab- 
lished for  all  parts  of  the  plane. 

Referring  now  to  equations  (24),  let  us  write  them  in  the  form 

dx 

'^    +oj'  +  oy-y'  +  o/''+ . 

dx 


-\-  oy"  ~ ' 

=  0, 

+  oy"  - ' 

=  0, 

+  oj'«-' 

=  0, 

(36)^ 


dx 


^''    ^  +  0/  +  oy'  +  oy"  +  oy'"  -j-  .  .  .  -f  07"  -  ^  —  j"  - '  =  o^ 


dx 
dy"- 


f  P>:y  +  Pn  -  .y'  +  /« -  ^y+Pn  -  ,y"'  + 


dx 

+  P.y"-'  +  P.y"-'  =  oJ 

The  adjunct  system  to  this  is: 
dx 


(37) 


-^^--  Y-^oY'+oY"-\-oY"'  +  .  .  .  +/,_,F«-'  =0,, 
dx 

-  '^-  -]-oY-Y'+oY"  +  oV"'-\-  .  .  .  +/„_,F«-'  =0, 
dx 


( 


-^^-f  oF+oF'+oF"  +  or"+  .  .  .-]-p,Y"-^  =0. 
dx 


GENERAL   PROPERTIES. 


17 


Multiplying  equations  (36)  by  F,  F',  .  .  .  ,  F"-%  respectively, 
and  equations  (37)  by  —  y,  —  y\  •  •  •  ,  —  J""*',  respectively,  and 
adding,  we  have 


or 

(38) 


ii^yy-^yy-v 


yy+yy  + 


_|_  y«  -  I  ^K  - 1^ 


o, 


+    Y"-^yn-^    ^    C, 


where  dT  is  a  constant.  This  result  has  already  been  obtained  in  the 
general  case  of  a  system  of  simultaneous  linear  differential  equations 
of  the  first  order.  If  we  omit  the  terms  in  (36)  and  (37)  whose  co- 
efficients are  zero,  these  two  systems  may  be  written  in  the  forms : 


(39) 


dy"  -  ' 

~  d7~ 

dy"  -  ^ 

dx 


dy 
dx 


-  y 


=  o 


and,  for  the  adjunct  system, 
dV" 


(40) 


dx 
dV"  -  - 


dx 


dY 
dx 


-  -\-  />,V"  -  '  —  Y"  -  ^  =  o, 

+  /,  F«  -  ^  -  F«  -  3    r:=   o, 


+A5^ 


o. 


Form  now  the  (;z  —  i)^'  derivative  of  the  first  of  these  equations, 
the  («  —  2)°''  derivative  of  the  second  equation,  etc.,  and  subtract 
the  sum  of  all  the  equations  of  odd  order  from  the  sum  of  those  of 
even  order ;  as  a  result  we  have 


1 8  LINEAR  DIFFERENTIAL   EQUATIONS. 

+  (-i)«/„F«-'  =  o. 
Replacing  V"-^  by  M,  this  is 

(*^>  -z;:^-^:?r^(^'^)  +  ?.-^.<^"^)-  •  •  •  +(-.)'A^/=o. 

This  equation  is  the  adjunct  equation  to 

(43)  ^^  +  A  ^rr  +  A  ^—  +  .  .  ..  +  AJ  =  o. 

The  meaning  of  (42)  is  easily  found.  Suppose  we  multiply  (43) 
by  the  indeterminate  function  AI  and  then  integrate  by  parts ;  we 
have  thus,  indicating  differential  coefficients  by  accents, 

(44)  My"  -  '  —  M'y"  -  ^  +  J/'>«  -  3_  .  .  .  -f  .  .  . 

+  (—  i)y>M\  dx  —  const. 
If  71/ is  a  solution  of  the  differential  equation 

(45)  M>^  -  {p,MY  -  '  +  {p,MY  -'-  ■  •  •  +  (-  OT'^  ^^  =  o, 

the  integral  in  (44)  will  vanish,  and  we  shall  have  a  linear  differential 
equation  of  order  «—  i,  containing  an  arbitrary  constant,  for  the 
determination  of  y.  If  we  know  k  solutions  M,,M^,  .  .  .  ,  M^  of 
(44)  we  shall  obtain,  on  writing  Af  =  M, ,  M=M,,  .  .  .  ,  M  =  Jh , 
successively,  k  linear  equations  of  order  n  —  i  in  y.  Eliminating 
between  these  equations  the  derivatives  y"  -  \  y"  -  ^  .  .  .  ,  y"  '*+', 
we  shall  have  for  the  determination  of  y  an  equation  of  order  71  —  k 
containing  k  arbitrary  constants.  It  is  clear  that  if  (42)  is  the  ad- 
junct equation  to  (43),  then  (43)  is  the  adjumct  equation  to  (42) ;  and, 


i 


GENERAL  PROPERTIES.  1 9 

further,  that  the  adjunct  equation,  say  ^  =  o,  to  a  given  equation, 
^  =  O,  is  simply  one  whose  integrals  are  multipliers  oi  A  =  O ;  that 
is,  supposing  B  =  o  and  A  =  O  to  be  each  of  order  n,  then  multiply- 
ing A  (or  B)  by  an  integral  of  B  (or  A)  and  performing  a  quadrature, 
the  equation  obtained  for  determining  the  unknown  function  in  A 
(or  B)  will  be  of  order  n  —  i,  and  will  involve  one  arbitrary  constant. 
The  subject  of  adjunct  systems  of  equations  will  be  resumed  in 
another  chapter. 

The  question  of  the  transformation  of  linear  differential  equations 
and  the  resulting  theory  of  the  invariants  of  such  equations  will  not 
be  dealt  with  in  the  present  volume,  but  a  few  remarks  may  be  made 
here  on  the  subject.  Suppose  we  have  given  a  linear  differential 
equation, 

p/'y  +  p/—l+p/—l+.   .   .+P,y  =  o. 

dx"  dx"  -  •  dx"  -^  -^ 

This  equation  can  be  transformed  in   two  different  ways,  so  that 

after  each  transformation  it  shall  retain  its  original  form.     We  may 

first  change  the   independent  variable  by  a    relation    of    the  form 

X  ^  f[t),  and  then  after  effecting  this  transformation  we  may  change 

the  unknown  function  jy  by  a.  relation  of  the  form  j  =  0  (^t)s.     The 

different  transforms  of  the  given  equation  obtained  by  giving  to  f{t) 

and  <p{t)  all  possible  forms  may  be  considered  as  belonging  to  one 

and  the  same  class.     Thus  all  differential  equations  of  the  second 

order  form  but  one  class,  "and  they  are  all  reducible  to  a  unique 

d'^n 
type,  say  -—  -j-  //^  =  O ;    but  we  do  not   know  how,  by  means  of 

simple  quadratures,  to  actually  make  this  reduction,  nor,  having 
given  two  equations  of  the  second  order,  do  we  know  how  to  find 
the  transformations  which  will  change  one  into  the  other.  The  cir- 
cumstances are  entirely  different,  however,  in  the  cases  of  equations 
of  the  third  or  higher  orders.  We  will  here  only  consider  briefly  the 
case  of  an  equation  of  the  third  order,  and  show  the  existence  of  in- 
variants in  this  case.     Suppose  the  equation  to  be 


20  LINEAR  DIFFERENTIAL   EQUATIONS. 

Transform  first  by  making  x  =  f{t) ;  writing 


(47) 


ZP'  = 


\0' 


R'  = 


dx  dx  ^dx' 


(£) 

> 

d't  .    ^d't 

ax                ax 

+  3^1 

\dxl 

R 

m 

> 

we  have 
(48) 


^^  +  3/'lf  +  3e'f  +  ^>  =  o. 


Now  makejj^  =:  0(^)-,  and  write 


3'f +  3/". 


(49) 


3-7;7  +  6/'^  +  3e<A 


i?„  = 


<p 


We  have,  as  the  result  of  this  second  transformation, 


Now,  from  the  first  of  (47)  and  the  first  of  (49)  we  find 


GENERAL  PROPERTIES.  21 


d(^ 

dH 

(51) 

Po 

= 

dt 
0 

+ 

dx-" 

dx 

and, 

after 

easy 

redi 

ictions, 

(52) 

e  - 

-// 

odt    - 

] 

dx 

— r-  ^ 

-/Pdx 

The  function  e~J'^^^  is  then,  relatively  to  the  given  differential 
equation,  a  true  invariant,  since,  after  the  transformations,  it  repro- 
duces itself  multiplied  by  a  factor  which  depends  only  on  the  trans- 
formations effected.  As  already  mentioned,  it  is  not  intended  here 
to  go  into  the  subject  of  these  invariants,  so  we  will  confine  our- 
selves to  the  mere  enunciation  of  another  invariant  of  (46).     Write 

(53)  /=4/"  +  6/.f  +  ^|'-6Pe-3f  +  2ie. 

and 

(54)       /.  =  6p;  +  6pf^- +^:^v  ep.a  -  3f  +  ^R. ; 

from  these  we  can  readily  find  the  identity 

The  function  /  is  therefore  an  invariant  of  (46.)    Combining  the  two 
invariants  so  found,  we  have  the  invariant 

(56)  J^e^fPd-I, 
which  gives  the  relation 

(57)  /.=/<P\t). 
If  we  consider  the  adjunct  equation 
/.ON                       d'^i          d'  ,  d 


22  LINEAR  DIFFERENTIAL  EQUATIONS. 

and  denote  by  /  and  J  the  two  invariants  (46),  and  by  /„  and  J^  the 
corresponding  invariants  of  the  adjunct  equation,  we  have 

P 

(59)  Io=-L  /o=-J. 

The  preceding  results  are  taken  from  a  paper  by  Laguerre  in 
the  Comptes  Rendiis  for  1879.  I"  another  volume  the  investi- 
gations of  Laguerre,  Halphen,  and  others  will  be  taken  up  from 
a  more  general  point  of  view,  and  as  full  an  account  of  their 
results  as  is  possible  will  be  given.  It  is,  however,  impossible  to 
give  a  reaWy  /u//  account  of  the  subject  within  the  limits  of  this 
treatise,  but  at  least  enough  will  be  done  to  enable  the  reader  to  con- 
sult with  profit  the  original  memoirs. 

In  what  precedes  we  have  assumed  that  the  difTerential  equa- 
tion possesses  an  integral,  but  it  is  obvious  that  this  fact  ought 
to  be  proved.  The  proof  of  the  existence  of  an  integral  in  the 
case  of  a  linear  differential  equation  is,  however,  only  a  particular 
case  of  the  proof  of  the  existence  of  an  integral  for  the  general 
form  of  an  ordinary  differential  equation  or  a  system  of  such  equa- 
tions, and  it  is  not  within  the  scope  of  this  treatise  to  give  this 
general  proof,  for  which  the  reader  is  referred  to  the  memoir  by 
Rriot  and  Bouquet  in  Cahier  36  of  \.\\q  Journal  dc  l Ecole  Polytech- 
niqiic  (and  also  in  the  treatise  TJu'orie  des  Fonctions  Elliptiqiics 
by  the  same  authors).  Jordan  in  his  Cours  d' Analyse,  vol.  iii., 
also  gives  this  general  proof.  In  connection  with  Jordan's  proof 
the  reader  is  advised  to  consult  a  paper  by  Picard  in  the  Bulletin  des 
Seienees  Matlieinatiques  for  1888  entitled  Snr  la  convergence  des 
series  repn'sentant  les  inte'grales  des  equations  diffe'rentielles. 

Fuchs  in  his  first  memoir  on  linear  differential  equations  in  vol. 
66  of  Crelle's  Journal  gives  the  special  form  of  Briot  and  Bouquet's 
proof  which  is  applicable  to  the  case  of  linear  differential  equations, 
but  that  will  not  be  reproduced  here.  Still  another  proof  in  the 
case  of  the  linear  differential  equations  which  we  are  about  to  study 
is  given  by  Frobenius  in  vol.  76  of  Crelle  ;  this  last  proof  will  be 
given  in  the  chapter  devoted  to  Frobenius's  method  for  integrating 
such  equations. 


I 


i 


CHAPTER  II. 

LINEAR  DIFFERENTIAL  EQUATIONS  WITH  CONSTANT  COEFFICIENTS. 

The  following  investigation  of  these  equations  is  due  principally 
to  Hermite,*  and  in  part  to  Darboux  and  Jordan. 

The  first  method  of  integrating  linear  differential  equations  with 
constant  coefficients  is  due  to  Euleivand  is  briefly  as  follows:  Writ- 
ing the  equation  in  the  form 

where  A^,  A^  .  .  .  ,  A^  are  constants,  Euler  makes  the  solution  of 
this  equation  depend  upon  the  solution  of  the  algebraic  equation 

(2)  F{a),     =  a''^  A,a"-^-{- A,a"-'-\- .  .  .  -\-A„  =  o; 

d'y 
in  which  a^  replaces  'i—\}  =1,2,  .  .   .  ii)\n  equation  (i).    As  is  well 

known,  this  equation  is  obtained  by  replacing  j  in  (i)  by  C^^,  and  after 
the  substitution  dropping  the  factor  e'^'',  which  of  course  does  not 
vanish. 

Equation  (2),  that  is  Fi^oc)  =  o,  is  called  by  Cauchy  the  c/iar- 
acteristic  equation  of  the  given  differential  equation.  The  details 
of  Euler's  method  are  well  known  to  all  students  of  the  elementary 
theory  of  linear  differential  equations ;  but  Cauchy's  method,  which 
we  now  proceed  to  develop  and  which  is  based  upon  a  knowledge 
of  the  characteristic  equation,  is  not  so  well  known,  at  least  not  to 
English  readers.  Let  ^(«:)  denote  an  arbitrary  polynomial  in  a,  con- 
taining therefore  only  positive  powers  of  oc ;  we  will  now  consider 
the  integral 

(3)  y  =  J  -^x^y-^^' 

*  Equations  differentielles  lineaires,  par  M.  Ch.  Hermite:  Bulletin  des  Sciences 

Maihematiques.     1879. 

23 


24 


LINEAR  DIFFERENTIAL  EQUATIONS. 


where  the  contour  of  integration  is  any  closed  curve.  Suppose  first 
that  this  contour  contains  no  pole  of  the  function 

that  is,  contains  no  root  of  the  characteristic  equation /^(o')  =  o ; 
then  by  Cauchy's  theorem  the  integral  in  (3)  is  zero  and  we  have  the 
known,  but  useless,  integral  j  =  o.  In  order  to  obtain  effective 
solutions  of  the  differential  equation  we  must  then  draw  the  contour 
of  integration  in  such  a  way  that  it  shall  contain  one  or  more  poles 
of  the  function  integrated.  It  is  easy  to  verify  that  the  value  of  y 
so  obtained  is  an  integral.     We  have 


(4) 


y 


f 


dy  re°-^an{a) 


da, 


dx 


F{a) 


d'y_  _     re'^-aUI{a) 
d^'  -  J   ~~Ft 


F{a) 
d"y  rc°-'^a"II{a) 

fivn  J 


da, 
da. 


dx 


F{a) 


da, 


the  integrations  in  each  case  being  taken  round  the  same  closed 
contour  containing  one  or  more  of  the  poles  of 


F{a) 


that  is,  one  or  more  of  the  roots  of  F{a)  ==  o.  The  expressions  (4) 
substituted  in  equation  (i)  give  for  the  first  member  of  this  equation 
the  form 

re^^nia) 


F{a) 


or,  from  (2), 


fe'^-'Il{a)da ; 


CONSTANT  COEFFICIENTS.  2$ 

but,  since  n{(x)  is  a  polynomial  in  a  containing  only  positive  powers 
of  a,  we  have 

fe°-'''n.(a)da  =  o 

for  every  possible  closed  path  of  integration.     It  follows  then  that 


/'- 


F{a) 


-doc 


is  always  an  integral  of  the  given  differential  equation,  no  matter 
"vvhat  be  the  contour  of  integration.  A  remark  must  be  made 
here  concerning  the  polynomial  n{a),  which  has  been  assumed 
■of  any  degree  whatever.  This  assumption  might  lead  one  to 
think  that  any  number  of  arbitrary  constants  might  appear  in  the 
general  integral  of  the  given  equation ;  it  is  easy  to  see,  however, 
that  the  number  of  such  arbitrary  constants  can  never  exceed  n,  the 
■order  of  the  differential  equation.  The  degree  of  F(a)  is  of  course 
;/.  Suppose  now  that  n{a)  is  of  a  degree  greater  than  fi;  then  we 
can  write 

F{a:)  ^   '         F{a) 

Avhere  0{a)  is  a  polynomial  and  W(^a)  is  also  a  polynomial,  but  one 
whose  degree  is  less  than  n.     Form  now  the  integral 


^      F{a)         '         '^  ^    '         '  '^      F{a) 


da. 


TKe  first  term  on  the  right-hand  side  of  this  equation  is,  by  Cauchy's 
theorem,  equal  to  zero,  whatever  be  the  contour  of  integration,  and 
so  the  equation  reduces  to 

nc^-^'IIia)  Pe°-'^W(a) 

/  ~cr\~da  =    I  —  da  ; 

where  W{a)  contains  at  most  ;/  arbitrary  constants,  since  its  degree 
is  at  most  n  —  i.  We  proceed  now  to  determine  the  explicit  form 
>of  the  integral 

--'-71(a) 


r  =    /  — r^. — r-da. 
•^         ^       F(  a\ 


26  LINEAR  DIFFERENTIAL  EQUATIONS. 

Denote  by  S  the  sum  of  the  residues  of  the  function 

which  correspond  to  the  roots  of  F{a)   lying  inside  the  contour  of 
integration.     The  integral 


/ 


da 


F{a) 


has  now  the  value  27tiS. 

We  will  first  suppose  that  the  characteristic  equation  F{a)  =  o 

n(a)  . 
has  no  multiple  roots,  and  then  decompose  the  function  -pr^  mto 

simple  elements.  As  we  have  seen,  the  degree  of  n[a)  may  be  sup- 
posed less  than  that  of  F{a),  and  so  we  shall  have  as  the  result  of 
the  decomposition 

(6)  ^)  =  _^.  +  ^L_+...+^._ 

^  ^  F{a)        a  —  a^    ^    a  —  a^  '    a  —  a„ 

e^^IJia)    ,  .  ,     ,       ,  . 

In  the  function  — „.  \     change  a  mto  «,  -f-  /^;  then,  smce 
r\C()  ° 


a  —  rt, 
I 


gives  only  one  term  containing  y,  we  have 

(7)  —F(^iir  =  '■"  L'  +  T  +  7:J  +  •  •  ■  J 


X 


c. 


'  +  A  +  A/^  +  A/^'+.  •  •  ]• 


The  residue  in  this  case  is  thus  =  C/"-''.    We  have  then  as  a  first 
solution  of  the  differential  equation  27iiC^e''^^  obtained  by  integrating 


CONSTANT  COEFFICIENTS.  2y 

round  a  contour  which  contains  only  the  single  root  a^  of  F{a)  =:  o,, 
that  is,  only  the  single  pole  «,  of  the  function 

In  general  the  contour  of  integration  contains  any  number  of 
poles  of  —  ,  ,  and  consequently  the  general  integral  of  the  equa- 
tion will  be  of  the  form 

(8)  y  =  C.e"^''  +  (T/'^a-^  +  .  .  .  +  C„e'"^ ; 

where  C^,  C^,  .  .  .  ,  C„  are  constants  any  or  all  of  which  may  be  zero 
according  to  the  path  of  integration,  and  where  a^,  a^,  .  .  .  ,  a„  are 
the  roots  of  the  characteristic  equation  F{a)  =  o.  The  factor  2m  is 
of  course  supposed  to  be  contained  in  the  constants  C^,  C^,  .  .  . ,  C„^ 
Suppose  now  that  F{a)  =  o  has  multiple  roots,  and  let 

(9)  F{a)  =  {a  —  a^Yi  +  '{a  —  a„y^  +  ^  .   .  .   (or— «^)^s  +  '. 
We  have  now  for  the  formula  of  decomposition 

c  c 


+ 

^  {a  —  «,)^  +  I  ^  (nr  —  a^Y^  + 1  ^  •   °   ■ 

Changing  a  into  a^  -\-  h  gives,  taking  account   only  of  the  negative 
powers  of  h, 

^^  ^^  f7[^      I      AN  —   77  +  "v:^  +    •    •    •    + 


F{a^-\-  k)        k    '     -/^-    '     •  •  •    1    >^^i+x' 


28 

also 

(12)       ^-''<«i  +  '^)  =  e"- 


LINEAR  DIFFERENTIAL   EQUATIONS. 


hx        Jl^x^  Ji^vK       -\ 

I    '  1,2    '  '   1,2, 


.K 


The    residue    corresponding   to  ar  =  a^,  that   is,  the    coefficient    of 


,,-r(aj  -(-A) 


is  found  by  multiplying  together  the  corresponding  terms  in  the 
right-hand  members  of  (ii)  and  (12).  We  find  thus  for  this  residue, 
and  consequently  for  an  integral  of  the  difTerential  equation,  the 
expression 


27rz^^i' 


.   '^     I     ^"  •  '^  1,2,   .  .  .  ,  A,.. 


(Of' course  the  residue  alone  does  not  contain  the  factor  27ri.)     The 
general  integral  of  the  given  equation  is  now  of  the  form 


(13) 


J  =  ^''■-''(?A,  +  ^''^^Qk,  +  .   .  .  +  r«3-^;,^  ; 


where  Qx,  is  a  polynomial  in  x  of  degree  A^.     Since 

(A.+  i)  +  (A,+  i)  +  .  .  .  +  (A,+  1)  =  ;., 

we  see  that  the  general  solution  contains  7i  arbitrary  constants. 

It  is  often  desirable  to  determine  the  arbitrary  constants  in 
the  general  integral  in  such  a  way  that  j'  and  its  first  ;z  —  i  deriva- 
tives shall  have,  for  a  given  value  of  x,  certain  specified  values.  In 
the  case  where  the  characteristic  equation  -F{(x)  =  O  has  all  of  its 
roots  different,  this  determination  is  easily  seen  to  depend  upon  the 
solution  of  a  system  of  Ji  linear  algebraic  equations.  Suppose,  in 
fact,  that 


J  =  C/'^i''  +  ^^Z"*''  +  •  •  •  +  C„e''n^ 


CONSTANT  COEFFICIENTS.  29 

is,  in  the  case  of  unequal  roots  of  F{oc)  =  o,  the  general  integral  of 
the  given  differential  equation,  and  suppose  ;f  =  o  to  be  the  particu- 
lar value  of  X  for  which  y,  y' ,  y" ,  .  .  . ,  y"  ~  ^  are  to  have  the  values 
J'o '  J'o'j  y/'y  •  •  •  >  fo"  ~  ' ;  then  for  the  determination  of  the  constants 
C^,  C^,  .  .  . ,  Cn  we  have  the  system  of  equations 


(H) 


C,a^        +  C«„         +  .   .  .  +  C„a„         =  y/, 
C,a,"  -  ^  +  C,a,"  -  I  4-  ...  +  C„a„"  -^  =y-- 


If,  however,  the  characteristic  equation  has  multiple  roots,  this 
method  cannot  be  easily  applied,  as  the  derivatives  of  y  are  more 
complicated  and  the  different  roots  do  not  enter  in  so  simple  a 
manner  in  the  equations  to  be  solved.  Cauchy  has  nevertheless 
given  a  very  simple  method  for  solving  the  problem,  which  applies 
equally  well  to  the  cases  of  simple  and  of  multiple  roots  of  the 
characteristic  equation.     This  method  we  now  proceed  to  develop. 

A  solution  of  the   differential  equation  has  been  given   in  the 
form 

I      Pe-^-nia) 

(15)  y  =  —     —d^^doc 


In  order  that  this  shall  be  the  general  integral  it  is  necessary  to 
draw  the  contour  of  integration  in  such  a  way  that  it  shall  contain 
all  of  the  roots  of  F{(x)  =  o.  The  contour  so  drawn  can  of  course 
be  expanded  indefinitely  without  altering  the  value  of  the  integral 
(15),  and  we  may  therefore  make  it  a  circle  with  centre  at  the  origin 
arid  of  indefinitely  large  radius.  It  is  now  required  to  determine 
the  constants  in  n{a)  in  such  a  way  that  for  ;ir  =  o  the  function 


V- 


271  vJ      F{a) 


da 


and  its  first  n  —  i  derivatives  shall  have  given  values,  say  jj/^ ,  //, 
y^"  '  ',  that  is,  so  that  the  equations 


30 


<i6) 


LINEAR  DIFFERENTIAL  EQUATIONS. 

I    nii{(x) 


2711 


an{a) 

F{a) 


27ti^ 


da  = 


^(--^^. 


F{a) 


I      fa"  -  'IKa) 
L  2711^         F{a)  •"' 


shall  be  satisfied. 


n{a) 


To  obtain  the  values  of  these  integrals  we  will  develop  -^-r  in 

descending  powers  of  a  ;  and  since  n{a)  is  in  general  of  degree 
n—  I,  the  first  term  of  the  development  will  be  of  degree  —  i,  and 
we  shall  have 


<i7) 


n{a) 
F{a) 


5  +  :7  +  ^^  +  ...  +  -^-  + 


Substituting   this  value   in   equations  (i6)  and   integrating   round   a 
circle   of  infinitely  great  radius,  we  know  that  we  need   only  take 

account  in  each  integral  of  the  term  in  - ;  we  have  then,  by  a  known 

formula, 


(i8) 


^0  =  Jo  . 

^i      =  y.' 


We  have  thus  found  the  coefificients  in  n{a)  of  all  the  terms  of 
degrees  equal  to  or  greater  than  —  n,  and  TI{a)  is  thus  completely 
determined  since  we  have  identically 


(19) 


n{a)  =  F{a)[l^^^-^,  +  .  .  .  +-^-_ 


I 


CONSTANT  COEFFICIENTS.  3 1 

and  since  lI{oc)  is  to  be  an  entire  polynomial ;  consequently,  since 
F{oc)  is  of  degree  n,  we  see  that  the  first  ti  terms  of  the  series  (17)  are 
alone  necessary  for  the  determination  of  the  polynomial  ^{pt),  and 
we  thus  have 

<20)     U{a)=y^     [^^_^_|_^^_^ar  +  yi„_3«r=  +  .   .   .  +  ««-'] 

+ 

We  have  thus  obtained  Tlia)  by  a  method  equally  applicable  to 
the  cases  of  simple  and  of  multiple  roots.  In  order  now  to  obtain 
the  explicit  form  oi  y,  it  is  only  necessary  to  calculate  by  the  ordi- 
nary elementary  algebraic  process  the  residues  of  the  known  func- 
tion 


F{a) 


As  a  simple  application  of  the  preceding  method  consider  the 
well-known  differential  equation 

^.-  +  ny  =  o. 

The  characteristic  equation  is  obviously 

or'  -|-  ;/=  =  o, 

and  has  for  roots  a  =  ±  in.  Suppose  now  that  for  ;»:  =  o  we  wish 
to  have  jj/  =  j^ ,  y  —  y^\  then,  by  (17),  the  polynomial  Uipi)  is  to  be 
determined  by  the  relation 

F{a)  ~"  or   "T    a'  ~r    ^3   ^  •  .  •  , 


32  LINEAR  DIFFERENTIAL  EQUATIONS. 

which  gives  on  multiplication  by  F{oi),  =  a^  -[-  ^^^^  and  disregarding; 
terms  of  the  product  containing  negative  powers  of  a, 

n{(x)  =  y.^  +  Jo'- 

We  have  now  to  calculate  the  residues  of 


F{a)     '  a'  +  n' 


ax 


For  a  rootA-a  the  residue  is  =  ~t^, — t"  or  "lj^o  +  ~    K""';  for  a  root 
'  I^  \oc)         2V-^     '    ex  I 

—  a  the  residue  is     I  j„ ^jr"-^.    The  sum  of  these  two  residues  is 


1-1/  ' 


|j/„(r-  +^---)  +  ij, 


Making  now  a  =  zV/,  we  have  readily  for  the  general  integral  sought 


sm  nx 
y  =  y,  cos  nx  +  y,  ——-. 


From  the  form  of  the  differential  equation  it  is  clear  that,  if 
y  =  <p[x)  is  a  solution,  then  y^  =  (p{x  -f-  0  ^'^^  ^^^o  be  a  solution, 
c  denoting  an  arbitrary  constant.    If  then  we  write 

sin  n{x  —  c) 
y  =y,  cos  n{x  -  c) -^  y, , 


y  and  y'  will  take  the  values  y„  and  y/  for  x  =  c.     A  number  of 
consequences  arising  from  the  general  integral  when  x  is  real  might! 
be  given  ;  but   it   is  not  necessary  to  give  them,  as  the  reader  is  of 
course  supposed  to  be  familiar  with  the  ordinary  elementary  theory^ 
of  differential  equations. 

We  will  now  briefly  consider  the  case  where  the  right-hand  mem- 


CONSTANT   COEFFICIENTS.  33 

ber  of  the  equation  is  not  zero  but  a  function  of  the  independent 
variable ;  say  the  equation  is 

d^v  d"  ~  'r  d'^~  ^v 

(^■)      ^  +  ^'iS^T  +  ^.557/7  +  ...  +  A„y=  f(.). 


f{x)  denoting  an  arbitrary  function  of  x.  The  following  method  for 
integrating  this  equation  is  due  to  Cauchy  and  is  given  by  Darboux 
in  a  short  paper  immediately  following  Hermite's  paper  above 
referred  to.     We  form  first  a  solution  of  the  equation,  say 

y  =  ^x,  t), 

which,  with  its  first  n  —  2  derivatives,  vanishes  for  x  ^=  t,  and  for 
the  same  value  of  x  has  its  {n  —  i)^"^  derivative  equal  to  f{t).  Such 
a  function  being  formed,  we  can  easily  verify  that 

(22)  y=/^^^{x,t)dt, 

where  x^  is  an  arbitrary  constant,  is  a  particular  solution  of  the  given 
differential  equation.  The  formation  of  ^  {x,  t)  is  quite  simple. 
From  what  precedes  we  know  that  the  integral 

I      ne'^^da 
271  i J   F{oc) 

(F{a)  having  the  same  meaning  as  before)  taken  round  the  circum- 
ference of  a  circle  with  very  great  radius  is  a  solution  of  the  differ- 
ential equation  when  its  second  member  is  zero,  and  that  this 
solution,  together  with  its  first  n  —  2  derivatives,  vanishes  for  x  z=  o, 
and,  finally,  that  its  u  —  i  .derivative  is,  for  this  same  value  of  x,  equal 
to  unity.  If  now  in  this  integral  we  change  x  into  x  —  t  and 
jmultiply  the  integral  by  F{t),  we  obviously  have  the  function  sought 
igiven  by  the  equation 

I    X  f(r)  nc'^^-^-^'^da 

.23)  ^^:,,t)=-^^^^-r  T-—^^, 

^         2m  J  R     F[a)      ' 


34 


LINEAR  DIFFERENTIAL   EQUATIONS. 


the  integration  extending  round  the  circumference  of  a  circle  of  in- 
definitely large  radius  R.     The  double  integral 


(24) 


^=.-^/>)'''^- 


e°-^^-*)da 


F{a) 


is,  as  we  shall  now  show,  a  particular  solution  of  the  given  differ- 
ential equation.     DifYerentiating  (24),  we  have 


(25) 


dY _  fix)   p  da  I      r""  rf -.7    pe'^'^'-' -  *^ ada 

dx  ~  27iiJj^F{a)  '    27iiJ_^/^'    Jr       F{a) 


da 


P  da 
the  integral   /  -7^  -;  is  known  to  be  zero,  and  therefo 
^       ^RF{a) 

(26)  T   =— -•/  At)dt         ., 

^       ^  dx  27tlJ^^  -"^  '      Jr         I\a 


t^aU  -  t)  ()i(loi 


Now  we  know  that  if/  <  ;/  —  i,  all  the  integrals 

r*at'da 

must  vanish,  and  consequently  by  differentiation  of  (26)  we  get 


(27) 


d^Y 
dx^ 


so  long  as  the  inequality  p  <Cn  is  satisfied.     Suppose  />  =  n  ~  i  ; 
then  from  this  last  equation  we  derive 


(28) 


d "  Y      fix)  pa  "-\da         I 


a,(.r  -  t)ai>tda 


dx"         27ti 
and  since 


pa"-''  da    ^      I      A''  ^ ,  ,  ,    /^r'-''  -  '^a' 


J  Pa"  ~  ^da 


\ 


CONSTANT  COEFFICIENT^,. 


35 


we  have  finally 

(39)       -^  =  /w + 3^/  fcy^l—F^sj 


^go.{x  -  t)„n^^ 


Substituting  the  above  values  of    Fand  its  derivatives  in  the  differ- 
•ential  equation,  we  see  that  the  equation  is  satisfied  by 

^  2  7TlJ_,/^'     Jr      F{a) 


since 


(30) 


F{oi)  =  a*'  -^  A^  a"--"  -{-  .  .  .  -\-  A, 
Referring  again  to  (24),  let  us  write 

I      /'^"(■^  -  *^da 


R 


I     pe^^""  -  ''a 
^  '  ~  271  i J j^     F{(x) 


then  R{t)  is  the  sum  of  the  residues  relative  to  all  the  roots  of 
F{a)  z=  o.  Suppose  -^r^  to  be  decomposed  into  simple  fractions, 
and  write 


F{a!)  i  a  —  a       {a  —  a)' 


4-  I      ^^-'    )   . 

I"  '  '  •  "^  (a  —  «)^  J   ' 


we  have  also 


i(-r  -  i)    —    ^^(-r  -  t) 


\  _^  (^  -  ^)  (^  -  0  ,         -1 


and    consequently   the    residue    corresponding    to    the   root   a   of 
F{a)  =  o  is 


L  I  '    1 .  2  ..../—  I  J 


36  LINEAR  DIFFERENTIAL  EQUATIONS. 

We  have  therefore  for  R{t)  the  value 

(3.)   ^»  =  ^.<.-.|^.  +  MLnl)  +  ...+5^.fc^)^-|, 

and,  finally,  for  Y  the  value 

(32)  Y=  ff{t)R{t)dL 

We  will  now  give  a  brief  account,  taken  from  Jordan,  of  systems 
of  linear  differential  equations  with  constant  coefHcients.  We  have 
seen  that  any  such  system  can  be  reduced  to  an  equivalent  system 
of  the  first  order.  Supposing  this  reduction  to  have  been  made,  we 
will  assume 


(33) 


^   +  ^n  J:   +   «nJ.   + 


+  <^zsys  =  o, 

+  a^s^s    =    O, 


^  +  ^s^y,  +  as,y,  + 


+  ^ss  ys  =  o, 


as  our  system  of  equations.     Denote  by  ^  the  characteristic  deter- 
minant 


asi,  a,^,  .  .  . ,     ass  +  « 

and  let  ^,,  ,  A^^ ,  .  .  . ,  ^„  denote  the  minors 

d^       dA  dA 

da'     da'     -  •  '■>     ^^ 


CONSTANT  COEFFICIENTS. 


2,7 


of  this  determinant.     Substitute  now  in  equations  (33)  the  following 
expressions  : 


(34) 


I     r^A  +  A  J,  +  .  .  .  +  AsA 


y. 


ys 


27tvJ                            A 

I     PAJ^^AJ,^. 

.  .  +  A,A, 

2niJ                             A 

I    cA.sOA-  A..A  +  .  • 

.  +  A,A 

27tl'- 


'e'^^da, 


where  0^,  d„,  .  .  . ,  6^  are  functions  of  a,  and  where  the  integration 
extends  around  an  arbitrary  closed  contour.  From  these  equations 
"we  obtain  by  differentiation 


<35) 


dx        2nid  A  ' 

dy..         I     rAJ,  +  AJ^  +  .  .  .  +  A^As       , 
dx  =  i:nV  "  '-      "A- ^  ^' 


dx        27ii^  A 


e'^^da. 


Substituting  in  the  first  of  equations  (33),  and  observing  that 


(36) 


f  (^„  +  a)A^^  +  a,,A,,  + 
I   («„  +  a)A._,  +  a,,A,,  + 


I  (^n  +  oc)As^  +  a,,A,^  +  . 
we  have  as  the  result 


.  +  a,sA,s  =  A, 
•  ~r  ^isA-2s  =  o, 

.   +  ^isAss    =   O, 


.  /  B^e°-^da. 


38 


LINEAR  DIFFERENTIAL   EQUATIONS. 


Making  the  corresponding  substitutions  in  the  remaining  equa- 
tions of  the  system  (33),  we  have  as  the  result  of  all  the  substitutions 
the  expressions 

(37)         .  I  Op'^^da, .  /  dp'^^da,     .  .  . ,     ;  /  d^e^^da. 

^^'^         2ni^      '  2ni^     '  '     2ni^ 

If  we  suppose  6^,  6^,  .  .  .  ,  6^  to  be  arbitrary  constants,  the  func- 
tions 6'/"-^',  .  .  . ,  0^e°-^  will  be  integral  functions,  and  the  integrals 
(37)  will  all  vanish,  and  therefore  the  expressions  in  (34)  will  be 
solutions  of  equations  (33),  and  these  solutions  will  contain  s  arbi- 
trary constants. 

Let  us  assume  that  we  have  chosen  a  circle  of  infinite  radius  as 
the  contour  of  integration.  Now,  the  initial  value  of  j/j  for  x  =  o 
will  be 


I     rA„«,  +  AJ,  +  .  .  .+A,A 


(38)  y:  =  j~/- 

but  we  have 

(39)  A  =  a'  -\-  Ba'  -'+... 
and 


da ; 


(40)  AJ,  +  AJ,-\-...-\-A,,t 


(To, 


^S   ,  ^S2    J 


=  e,a'  -^-i^  Dc 


a,,-\-  (y 


+ 


and  consequently 

140  J  -  a^  a'~^ 


i 


and 


(42) 


•^'  27Tr'    \a    ^    a  / 


CONSTANT  COEFFICIENTS. 


39 


We  can  find  in  like  manner  the  initial  values  of  j^ ,  jj ,  .  .  .,  and 
so  have  finally  for  these  values 


y:  -  ^. ,  y. 


y 


"=  ^„ 


ys  = 


The  solution  which  we  have  now  found  is  the  general  integral, 
since  by  properly  choosing  the  constants  ^i ,  ^2  >  •  •  •  >  ^s  we  can  give 
yi  y  y-if  '  •  •  y  ys  arbitrary  initial  values.  The  values  of  the  integrals 
(34)  are  easily  found  ;  each  one  is  in  fact  equal  to  the  sum  of  the 
residues  corresponding  to  the  roots  of  J  =  o  of  each  of  the  func- 
tions which  is  to  be  integrated.  Consider  the  integral  j, ,  and  let  «, 
denote  a  root  of  J  =  o  whose  order  of  multiplicity  is  =  /^ ;  then 


(43) 


AJ.-{-A„J,-\-.  .  .    -{-AsA 


F^ 


{a  -  ay 
'    {^  -  «.) 


+  .   . 


+  ^0  + 


where  F^  ,  .  .  . ,  F^  are  linear  functions  of  the  constants  6^,  0^, 
0^.     Again,  we  have 


(44) 


=L     ^Oji 


I  +  (a  -  a,)x  -\-{a-  a)'^^  + 


Forming   the    product    of   (43)  and    (44),  we  have  for  the  sought 
residue  the  expression 


(45) 


F,  +  F,^'  +  '  -  ■  +  F. 


x^- 


I  .  2  ....(//-  i)J 


e"-^' 


That  part  of  the  value  of  jj/,  which  corresponds  to  the  root  a^  is 
then  of  the  form  Qe''^'^ ,  where  Q  is  a  polynomial  in  x,  in  general  of 
degree  /^—  i.  This  degree  will,  however,  be  lowered  to  fj.  —  k  —  i 
if  the  minors  A,^,A^^,.  .  . ,  A^^  are  all  divisible  by  (a  —  «,)*,  since 
in  this  case  we  shall  obviously  have 


iV  =  •  •  .  =  ^M  -  «  +  I  =  o 


40 


LINEAR  DIFFERENTIAL   EQUATIONS. 


We  will  now  consider  briefly  the  case  when  the  right-hand  mem- 
bers of  the  system  of  equations  are  not  zero,  and  write  the  equations 
in  the  form 


(46) 


dx 


+    «uX    +    «12j.    + 


dy. 


dx 


V    +   ^21  J:    +   ^.27. 


.  +  a.sys  =  fix\ 
.  +  a,sys  =  Ai'V), 


dy, 
-  dx 


+  a,,y,  +  a,,y^  -|-  .  .  .  -^  a,,y,  =  fix). 


Formulae  (34)  will  give  a  particular  solution  of  this  system  if  we 
determine  the  functions  6^,  B^,  .  .  .  ,  6^  and  the  contour  of  inte- 
gration in  such  a  way  that  the  equations 


(47) 


.  /  e^e^-'da  =  f  , 

27ll'J      '  -^' 


shall  be  satisfied.  This  determination  is  easily  arrived  at  if  /,,/,, 
.  .  .  ,  /^  are  of  the  form  gf^-%  where  Q  is  a.  polynomial.  Suppose, 
in  fact,  that 


(48) 


/  =  (^0  +  ^>^  +  •  .  .  +  ^.«^'")^'-^ ; 


then  in  order  to  satisfy  the  first  of  equations  (47)  we  have  only  to 
make 


(49) 


CONSTANT  COEFFICIENTS.  4 1 

and  integrate  around  a  small  circle  containing  the  point  A.  We  see 
at  once  from  this  that  if  /i ,  /^ ,  •  •  •  ,  fs  are  polynomials  of  order 
1)1,  then  0^,  6^,  .  .  ,  ,  6^  will  be  sums  of  simple  fractions  containing 
powers  of  or  —  A  in  their  denominators  up  to  {a  —  A)'«  +  i. 

We  have  consequently,  fx  being  equal  to  zero,  or,  if  A  is  a  root  of 
A  ■=  o,  fx  being  the  order  of  multiplicity  of  this  root, 

/.  N     ^.A+^.A+.  ■  ^+A,^d,  _  G 

\50;  ^  -    ^^_^y„  +  ^+.    -1-     .     .     .     . 

"^  «  -  A'  "^  •  •  •  ' 
and  the  corresponding  value  oi  y^,  which  is  equal  to  the  residue  of 

A 
for  the  point  A,  will  be  of  the  form 

where  Z  is  a  polynomial  in  general  of  degree  in  -[-  /^,  but  will  be  of 
lower  degree  if  the  first  coefficients   G,  G^,   ...  vanish.     Similar 
results  will  of  course  be  obtained  for  ja ,  y^ ,   •  •  -,  Ji- 
lt is  well  known  that  every  equation  of  the  form 

<5i)     {ax  +  bY^2n  +  ^'(^^  +  ^y  -  '1^-^  +  .  .  .  +  ^«J/  =  o 

can  be  thrown  into  the  form  of  an  equation  with  constant  co- 
efficients.    Writing,  in  fact, 

ax  -{-  b  :=  (?', 
we  have 


dy  _  dy 

dx  dt^ 


42  LINEAR  DIFFERENTIAL  EQUATIONS. 

and,  in  general, 

(52)  ^3^=«^.-«P.; 

dy 
where  Pk  is  a  linear  function,  with  constant   coefficients,   of   -7  „ 

at 

d^y  d^y 

~j7ij  •  •   •  >  ~j7j^'     Assuming  (52)  to  be  true  iox  k  {k  =  i,  2,  .  .  .  ,  ),. 


it  is  readily  seen  to  be  true  for  k  -\-  i  ;  we  have  in  fact 


dx^+^  ~  dt 

=  a^  +  'c-'i-  ke-  ^' Pk  +  e-  '''^^ 

Substituting  these  values  in  (51),  we  have  for  the  determination  of 
jj/  as  a  function  of  ^  a  linear  differential  equation  with  constant 
coefficients.  If  the  corresponding  characteristic  equation  has  all  of 
its  roots  unequal,  the  general  integral  will  be  of  the  form 

(53)  y  =  cr^'  +  .  •  •  +  C^"'^' 

=  Ciax  +  by^  +  .  .  .  +  Clax  +  bf-  . 


i 


If  the  characteristic  equation  has   multiple  roots,  then   to  any  one 
of  them,  say  «, ,  there  will  correspond  a  solution  of  the  form 

(54)  .«4^+Q+  ...  +6^,-,/'^-'] 

^  {ax  +  by^iC^  C,  log  {ax  +  /;)  +  ...+  C^  _  Jog  '^  -  \ax  +  b)\ 

We  will  find  solutions  similar  to  these  in  the  more  general  class 
of  linear  differential  equations  which  we  shall  presently  study. 


I 


J 


CONSTANT  COEFFICIENTS. 


4S 


In  the  Bulletin  des  Sciences  MatJi^matiques  for  August,  i! 
M.  Ch.  Meray  gives  an  investigation  of  the  differential  equation  with 
constant  coefificients,  of  which  the  following  is  an  account : 

Consider  the  k  series 


{a) 


f   U{z)  =  u,  +  n^z  +  7t,s'  +  .  .  .  +  u,„s"'  +  .  .  .  , 
V{z)  =  V,  +  v,z  +  v,z^  +  .  .  .  +  v„,z"'  +  .  .  .  , 

•  > 
T{z)  =  4  +  t,z  +  t^z^  +  .  .  .  +  t„,z"^  +  .  .  .  , 


depending  upon  the  same  variable  z ;  we  will  say  that  these  series: 
are  co-recurrent  if,  for  all  values  of  the  index  ;//,  we  have  the  k  rela- 
'tions  or  recurrences 


iP) 


•  +  lijm  =  O, 

•  +   /ijm     =   O, 


f,„  _|_  I   +  <^A-«;«  +   <^^^';«  +   .     .     .   4-  /l^t,„   =   O  ; 


of  which  the  coefficients  {a,  b,  .  .  .  ,  k)  are  arbitrarily  given  con- 
stants. The  quantities  {n,  v,  .  .  .  ,  t)  are  now  all  known  when  we 
know  (?^„ ,  7'(,  ,  .  .  .  ,  /„).  The  summation  of  these  series  is  easily 
effected  by  aid  of  the  theorem  :  T/iese  series  are  convergent  for 
values  of  z  zvitJi  sufficiently  small  moduli,  and  writing 


\f) 


I  -f-  a,z, 


Kz 
h  z 


a^z,      I  +  b^z,  .  .  .  , 
a^z,  bkZ,  .  .  .  ,     I  -\-  /iiS 

{F(z)  a  polynomial  in  z  of  degree  k),  also  writing 
[  Aiz\     Biz),     .  .  .  ,     Hiz), 

A,{z),    B,{z),    .  .  .  ,     H,{z) 


=  F{^\ 


{d) 


44 


LINEAR  DIFFERENTIAL   EQUATIONS. 


to  denote  the  minors  of  F  corresponding  to  the  like-placed  elemeiits  in 
Fiz),  we  have 


{e) 


U{z) 


F{z) 

u,Biz)  +  v,B,{z)  +  .  .  .  +  t,B,{z) 
F(z) 


r{z) 


,Hlz)  +  v^Hlz)  +  .  .  .  +  t,H,{z) 
F{z) 


The  numerators  in  these  are  obviously  polynomials  of  degree  k —  I. 
Admitting  for  the  moment  that  the  series  {a)  are  convergent,  we 
readily  find  that  their  sums  are  connected  by  the  k  simultaneous 
linear  equations 


(/)      ^ 


{  {I  +  a,z)U  +  b,zV  +...^h,zT  =u,, 

a,zU  +  (I  +  M  V  A-  .  .  .  +  h,zT  =  v, , 


a/,zU 


+  b,zV 


+  .  .  .  +  {i+h,z)T=  t. 


For,  adding  the  second  members  of  equations  {a)  after  multiplying 
them  respectively,  for  example,  by 

I  +  a^2,     b^z,     .  .  .  ,     h,z, 


we  see  that  the  term  which  is  independent  of  z  reduces  to  ?/„  and,  by 
the  first  of  recurrences  {b),  all  the  other  terms  reduce  to  zero  when  we 
make  m  successively  =  o,  i,  2,  .  .  .  ,  and  the  same  holds  for  equa- 
tions {/)  other  than  the  first  one.  The  solution  of  these  eqiiations 
gives  us  (<?).     Now,  since  we  have 


ig) 


F{z),     =  I  ^p^z^p,z'  +  .   .   .  -\-p,z^ 


a  polynomial  which  does  not  vanish  with  z,  we  know  by  elementary 
principles  of  the  theory  of  functions  that  the  rational  fractions  (e) 


CONSTANT  COEFFICIENTS.  45 

are  developable  by  Maclaurin's  theorem  for  values  of  z  for  which 
I  mod.  z  is  equal  to  or  less  than  the  least  modulus  of  the  roots  of 
\F{z)  =  o.  The  integral  series  obtained  by  these  developments  can- 
not differ  from  the  proposed  series  {ci)  since  they  satisfy  equations 
(/),  which  are  equivalent  to  the  recurrences  {b).  Each  of  these 
series  considered  separately  is  obviously  recurrent  in  the  ordinary 
sense  of  the  word. 

// 

0.{z\ 
\(Ji)  -^  =  zv,  +  z£;,^  +  .  .  .  +  w„,^-  +  .  .  . 

is  the  development  in  a  recurrent  series  of  a  rational  fraction  in  z^ 
whose  niunerator  fl{z)  is  of  degree  <  k,  zvhen  k  is  the  degree  of  the 
polynomial  F{z),  the  integral  series 

(,•)  ^^  _|_  3^  +  ^^-^-^  +  .  .  .  + "^ ^"^  +  .  .  . 

(^  ^  °    '      I        '1.2       '  '1.2 w  •'     ' 

:  is  convergent  for  all  the  values  of  z,  and,  representing  its  sum  by 


n{z) 
F{^) 


we  have  for  its  calculation 


s'^-^ni^- 


I.  When  the  rational  fraction  {h)  reduces  to  the  simple  fraction 


4^  LINEAR  DIFFERENTIAL  EQUATIONS. 

s^  being  any  constant  other  than  zero,  we  have  evidently 


^^(^  +  0  •  •  •  ($'  +  ^n  —  i) 


I   .    2 


in 


(I  -  ^,^)? 

^    1.2...!.(^-  l")  ^*^'''  +  ')  ('''  +  2)  •  •  .  (^/^  +  ^  -  O-y,'"^" 

We  deduce  successively,  and  without  difficulty. 


--^l 


7 C     .      =    ■ -y'^i^fl  +   l)  (^-'^  +  2)    .    .    .    (W  +  ^   —     l)— 


I         _j^^- 


-  d^--"    {sz)'"  +  ?  -  1  • 


^  —  i|      [  fl'^^ - ' 


_dz9  -  '  »«  +  ^  —  I 


I       r<3^?-'     ^?-' 


^-  I 


^-  I 


(S^^^  - '   dsi-"^ 


Now,  by  definition,  this  last  expression  is  simply  the  residue  of  the 
function  of  s, 

si  -  '^2 


with  respect  to  its  infinity  5,  of  order  of  multiplicity  q.     We  have 
then  definitely 


(/) 


I 1  _  cw   .     -y^  ~  '^"   _  „ 

(I  -  s,z)i  j  ~  .l^,^  ■    {s  -  s,y  ~  ,f,,' 


I        ^^ 


CONSTANT  COEFFICIENTS.  47 

which  proves  the  theorem  in  this  simple  case,  for  under  this  last 
X  the  coefficient  of  e"^  is 

s9  -"■  .  \ 


.M .  -  s!-^ 


11.  In  the  general  case  the  rational  fraction  {Ji)  can  by  decompo- 
sition be  thrown  into  the  form  of  a  certain  linear  homogeneous  func- 
tion of  simple  fractions  like  {k),  and  consequently  the  series  {i)  into 
the  form  of  a  linear  and  homogeneous  function  of  partial  series 
which  correspond  to  the  simple  fractions,  or  of  residues  expressing 
the  sums  of  the  partial  series,  as  in  equation  (/).  Collecting  all  these 
partial  residues  into  a  single  residue,  the  recomposition  of  the  simple 
fractions  analogous  to 


4)' 


will  evidently  put  the  product 


ni-'l  s'^-^d^^ 


F[~]  s^F 


(r 


under  the  sign  %,  which  completes  the  proof  of  our  theorem. 

III.  Our  reasoning  assumes  implicitly  that  F{z)  is  of  the  effec- 
tive degree  k ;  that  is,  that  pk  is  not  zero.  For  if  it  were  otherwise, 
the  rational  fraction  (//)  would  not  merely  be  resolved  into  simple  frac- 
tions, but  the  fractions  would  be  accompanied  by  integral  monomials 
in  z,  of  which  up  to  this  nothing  has  been  said.  We  can  neverthe- 
less show  (though  this  will  be  left  as  an  exercise  for  the  reader)  that 
the  generality  of  formula  {J)  will  not  be  infringed  upon  when  we 
take  account  of  the  simple,  or  multiple,  root  i'  =  o  which  is  now 
possessed  by  the  equation 


48  LINEAR  DIFFERENTIAL  EQUATIONS. 

and  of  the  evident  relation 


d        .  =  0^^  +  ^       s=o\sJ         s 


A  very  few  words  sufifice  now  for  the  treatment  of  our  principal 
question.     Consider  the  system  of  Hnear  differential  equations 


(m) 


du 


^-  +  a,u  J^  b,v  ^  .  .  .  -\-  h,t  -  O, 

dv 

-^  +  a^i  +  b^v  +  .  .  .  +  hj  =  o, 


dt 

-j^  +  a^n  -^  b  V  -\-  .  .  .  -\-  likt  =  O, 


the  coefficients  {a,  b,  .  .  .  ,  h)  being  arbitrary  constants.     Let 


in) 


ttk, 


bk,  .  .  .  ,     /h 


=  f{s)=s''+p,s^-^+  ...-^Pk; 


further,  let 


{0) 


be  the  table  of  principal  minors  of  this  determinant.     Calling  now 


iP) 


«0  ,  ^0  , 


,  t. 


CONSTANT  COEFFICIENTS. 
arbitrary  constants  {k  in  number),  and  making 


vis)  =  u,a^{s)  +  v^cxXs)  +  .   .   .  +  r,a'i{s), 
<P{s)  =  ic,i3ls)  +  v.f^ls)  +  .  .  •  +  tJh{s), 

^  r{s)  =  u^7]ls)  +  v.vls)   +  .  .  .  +  t,7j^{s) ; 


49 


Cauchy's  theorem  is  as  follows :    The  integrals  of  the   system  [m), 
which  for  x  ^^  x^  take  the  initial  values  {p),  are  given  by  the  formulcB 


ia) 


u  =  % 

s 

V   =    % 


~TfW~ 
~17is)]~ 


t  =  %- 


r(s)e'(-'-  -  -'-o) 


[/(^)] 


which  furnish  thus  the  general  integrals  of  the  given  equations. 

Making  x=^x^  in  equations  [in)  and  in  all  the  equations  deduced 
from  them  by  repeated  differentiations,  we  see  immediately  that  the 
initial  values  of  the  integrals  and  their  derivatives  of  all  orders,  viz., 


'^0  »       "i  >       '    '    •    t       ^im  >        •    •    •    > 
•^  0  J         *-  \  1        '     '     '     1        '■mi        '     •     '     ) 

^      ^0  >         ^1  »        •     •    •     >         t,„  ,        .     .    .     , 


are  connected  by  the  recurrences  (/;). 

It  follows  then  that  the  sums  of  the  co-recurrent  series  in  ^^  which 
have  these  initial  values  as  coefificients  are  given  by  formulae  (e),  and 
consequently  that  the  series  m  x  —  ,r„  are  deduced  from  these  last 
by  replacing  ^  by  ;ir  —  :f„  and  dividing  the  terms  in  (x  —  x^"'  by 
1.2 in;  that  is,  the  integrals  of  {in)  having  the  initial  values  (/) 


50  LINEAR  DIFFERENTIAL   EQUATIONS. 

are  obtained  by  applying  formula  (/).     These  integrals  are  then  the 
second  members  of  equations  (^),  for  we  have  evidently 


/(.)-.^/^Q, 


and  the  elements  of  the  table  {6)  are  just  what  the  corresponding 

elements  of  {d^  become  when,  after  changing  z  into  — ,  we  multiply 

(each  one  by  5*  ~  '. 

The  reader  is  referred  to  a  mernoir  by  J.  Collet  in  the  Annales  de 
TEcole  Norniale  Snpcrieiire  for  1888  for  another  treatment  of  linear 
•differential  equations  with  constant  coefficients. 


CHAPTER   III. 

THE   INTEGRALS   OF   THE   DIFFERENTIAL   EQUATION 
d"y   ,        d"-''y   ,        d"-^y    , 


The  coefficients  /i,  A'  •  •  •  >  /«  ^"^^  uniform  functions  of  x,  hav- 
ing only  poles  as  critical  points.  Starting  then  at  any  neutral  point, 
say  x^,  moving  along  any  path  whatever  (provided  of  course  it  does 
not  pass  through  a  critical  point),  and  returning  to  x^,  the  functions 
/  vary  continuously  and  return  at  the  end  of  the  path  to  their 
original  values.  Let  y^,  .  .  .  ■,  y,,  denote  a  system  of  fundamental 
integrals :  these  integrals  may  or  may  not  have  critical  points  (in  gen- 
eral of  course  they  have),  but  whatever  critical  points  they  have,  and 
o.f  whatever  description  they  are,  they  must  be  included  among  those 
■of  the  coefficients/.  When  the  variable  starts  from  ;r(,  and  travels 
by  any  path  back  to  x^,  the  equation  resumes  itS' original  form,  the 
coefficients/  having  returned  to  their  original  values;  the  functions 
J', ,  .  .  .  ,  jj/„  vary  continuously  along  this  path,  remaining  always 
integrals  of  the  equation ;  we  must  have  then,  obviously,  at  the  end 
of  the  path  that  j/j ,  .  .  .  ,  y„  have  been  changed  into  linear  func- 
tions of  themselves,  i.e.,  a  linear  substitution  has  been  imposed  on 
J/,,  .  .  .  ,  j/„ .  Denote  the  final  values  of  the  integrals  by  ^,,  .  .  ., 
Syn  ;  then  we  have 


(I) 


^J\    =    ^2iJi    +    •     •     •     +    ^2«J«, 

Sy„  =  c„,y^  +  ...-[-  c„„y„ ; 


51 


52 


LINEAR   DIFFERENTIAL    EQUATIONS. 


where  Cik  is  a  constant.      Or  we  may  say  that  the  integrals  jj//  have 
submitted  to  the  substitution 


(2) 


5  = 


y\ 


^uJ\     +     •     •    •    +  ^i«J« 


J'n]     c„,y^  + 


~\~  ^nnyn 


Before  proceeding  farther  with  the  study  of  the  integrals  it  will 
be  convenient  to  notice  briefly  the  case  of  equations  having  uniform 
singly  or  doubly  periodic  coefficients.    Suppose  that  in  the  equation 

the  coeflficients/j,  .  .  .  ,  /«  represent  singly  periodic  functions  of  x\ 
let  00  denote  the  period ;  then//(;r  -)-  oo)  ^p^ix).  If  then  we  change 
X  into  X  -\-  GO,  the  equation  retains  its  original  form.  In  the  investi- 
gation farther  on  of  the  integrals  of  this  equation  we  will  assume 
them  to  be  uniform  functions  of  x;  for  the  present  purpose,  how-  J 
ever,  no  such  assumption  is  necessary.  Suppose  /,(-*")'  f^i^)^  •  •  •  > 
/„{x)  a  system  of  fundamental  integrals  of  P  =  o;  i.e.,  a  linearly 
independent  system.  Since  in  travelling  from  the  point  x  to  the 
point  X  -{-  CO  the  equation  resumes  its  original  form,  and  since  the 
continuously  varying  functions  /i{x)  remain  always  integrals  of  the 
equation,  we  must  have 

'     /i(^'   +    ^)    =    «'n/,(-l')    +    «',./.  W    +   •     •     •    +    ^inM^\ 
/    N         ^       /.(-^   +    ^)    =    «'.i/,(^')    +    «'../.('!')    +    •     •     •     +    ^.«/«(-^)^ 

or  the  integrals/"/  have  been  submitted  to  the  linear  substitution 


5  = 


/;       ^n/   + 


/« ;    ««i/x  + 


+  «'i«/« 


+  ^««/« 


J 


Again,  suppose   the   coefficients  //  to  be   uniform  and  doubly  peri- 
odic functions  of  x,  and  let  oo  and  oo'  denote  the  periods ;  then 

p,{x  +  a?)  =  Xx),     Piix  +  oo'),  =  pi{x\ 


LINEAR   SUBSTITUTIONS. 


53 


Let  f^{x),  '  •  •  ,  fnix)  denote  a  system  of  fundamental  integrals  of 
the  equation  at  the  point  x.  If  now  we  travel  by  an  arbitrary 
path  from  x  \.o  x  A^  co,  the  integrals  fi  will  obviously  submit  to  the 
substitution 


5  = 


/;        «n/   + 


+  «'I«/« 


fn  ;        «'«i/,    +     .     .     .     +    0(„^f^ 


Again,  if  we  travel  from  x  to  x  -\-  oo' ,  the  integrals  will  submit  to 
the  substitution 


S'  = 


A  ;    «^'n/  + 


/«;    <./  + 


+  «'i«/« 


+  «''««/« 


Finally,  change  x  into  x  ~\-  go  -\-  co' ;  the  effect  of  this  change  is 
clearly  the  same  as  first  making  the  substitution  5  and  then  making 
the  substitution  S',  or  first  making  the  substitution  S'  and  following 
it  by  5.     We  have  therefore  the  relation 


<4) 


SS'  =  S'S. 


The  order  of  the  substitutions  is  the  order  in  which  they  are  written; 
^.g:,  S'S^  means  that  first  the  substitution  S'  is  made  and  then  the 
substitution  S^. 

Returning  now  to  our  original  equation,  P  =.  o,  let  j, ,  .  .  .  ,  y„ 
denote  a  system  of  fundamental  integrals ;   we  have  then 

d"v  d"^  ~  '1/ 


dx"    "T"  P\ix"  -  '  ~^ 


+  Aj.  =  o, 


54 


LINEAR  DIFFERENTIAL  EQUATIONS. 


Regarding  these  as  forming  a  system  of  linear  equations  in  /, ,  /,  ^ 
.  .  .  ,  pny  we  have 


(5)  Pi- 


d"  -  y^  d"-'  +  y,     dy,     d"-'-  y. 


d" -  y^  d" -  '  +  y^     d "y^     d"-'-  y^ 

dx"  -!'•••'   cix"  -  '■  + 1  '    dx"   '    dx"-  '-  '  ' 


d"  -  y„  d*' - '  +  y„    dy„    d"-^-  y„ 


dx" 


where 


(6) 


n  = 


dx''-'  +  '^   '    dx"  '   dx" 

d"  -  y,      d"  -  y, 


dx"-^  ' 

dx"--  '• 

•  ',J\ 

d"-y, 
dx" -  '  ' 

d"-y, 
dx" -  2  '  ■ 

",ya 

d"  -  y„ 

d"  -  y„ 

v.. 

dx"  -  '  '    dx"  -  2  '  *  • 


In  particular, 


(7)  A=- 


d"y,     d"-y, 
'dx"'   dx"-- ' 


d"y„    d"-y„ 


I  dx"  '   dx"--  '  ■  *  *' 
Then  we  have 
(8) 


}\ 


yn 


d"-y^    d"-y. 


dx"-"  '  dx"—  ' 


d"-y„    d"-y„ 

dx"--"  '  dx"--  ' 


.7i 


,yn 


D—  Ce-SP.dx.^ 


where  6"  is  a  constant  necessarily  different  from  zero,  since  the  in- 
tegrals jj ,  .  .  .  ,  yn  are  linearly  independent.  The  formula  for  pi 
may  be  written  for  brevity  as 


A=  - 


Di. 

d'' 


J 


LINE  A  R   SUBSTITU  TIONS. 


55 


where  Di  denotes  the  determinant  into  which  D  is  changed  when  its 
z'""    column    is    replaced    by  -j-^  ,  •  •  •  ,  ~r^'     The   result  of   going 

round  a  critical  point  is  to  change  the  integral  yi  into  Syj,  and, 
consequently,  to  change  the  determinants  D  and  Dj  into  /^D  and 
ADi,  where  A  denotes  the  determinant  of  the  substitution  S;  viz., 


(9) 


^n  '         ^. 


^«I  >        '«2  > 


•    •    >    ^I» 

•      •      >     ^2» 

•    •     >    ^nn 

Therefore,  after  going  round  the  critical  point,  we  have,  as  we  should, 


A  = 


AD 


D' 


It  is  of  course  to  be  noticed  that  the  determinant  A  is  not  equal  to 
zero  ;  for  if  it  were,  the  system  Syi ,  and  consequently  the  system  y^ , 
could  not  form  a  fundamental  system,  li y^  ,  .  .  .  ,y„  denote  a  system 
of  fundamental  integrals,  and  ^^ ,  ^^ »  •  •  •  ,  •s'a  linear  functions  with 
constant  coefificients  of /j ,  .  .  .  ,  j/a  ,  given  by 


(10) 


k  =  K 

Si  —  2  dk  yk ; 


then  ^j,  .  .  .  ,  ^A,  jj/a  +  i, 
determinant 


(II) 


A'  ^ 


,  jj/«  form  a  fundamental  system  if  the 


r       r  r 

^\\  '        '-'13  '     •     •     •     >     '-lA 


L-Ai  ,        Ca2  ) 


,G 


is  different  from  zero ;  in  fact,  it  is  obvious  that  if  A'  be  different 
from  zero,  there  can  exist  no  linear  relation  with  constant  coefficients 
between 


^1, 


,  -S'A,  JJ/a+i  , 


yn 


56 


LINEAR  DIFFERENTIAL  EQUATIONS. 


'•    Suppose  a  particular  integral  j,  (different  from  zero)  to  have  been 
found  for 

make  now 

y  =  yj-dx ; 

then  we  have  for  z  the  equation 

^«-'^  d"-^z   , 


where 


(12) 


dx"""    '   '^'^  dx^' 

n  dy^ 
y^dx 

n{n—\)d''y^      np^dy.^ 


^'=j;;^+^' 


;+A, 


^^        I  .  2  .  ji  ^a!';^^    '    ji    ^-^ 

n{7i  —  i)  ...(«  —  r  +  i)  ^'jf', 


^r  = 


I   .  2 


r.j, 


^'/r^ 


(;;  -  \){n  -  2)  .  .  .  {n  —  r +i)  rt^"  -  y. 


+A 

+  .--+A 


I  .  2  ....  (r  —  i)j/,  ^;i:'"- ' 

(;/— /^)(«  — /I'— i).  .  .{n  —  r-^i)d'--^y^ 


1.2 (r  -  k)y. 


dx' 


-J+-'--^Pr, 


Suppose  now  that  a  solution  z, ,  different  from  zero,  has  been  found 
{or  Q  =  o ;  substitute  as  before 

z  =  z^ftdx, 

and  we  have  for  /  an  equation  of  order  n  —  2.     This  process  may  be 
continued  until  we  arrive  at  an  equation  of  the  first  order  in,  say, 

ii) an  integral  of  which  is  w, .     We  have  now  as  integrals  of  P=  o 

the  following: 

J,,   y^=yjz4x.  y,=yjz^dxft,dx,  .  .  . 

y„  —  y  J z^dx f  t ^x  f  u^dx  .  .  .  fw^dx  ; 


LINEAR   SUBSTITUTIONS.  57 

and  these  constitute  a  fundamental  system  ;  i.e.,  we  can  have  no  such 
relation  as 

Cj,  +  C^y^  +  .  .  .  +  C,,y„  -  o 

unless  all  the  constant  coefficients  C  are  equal  to  zero.  For,  suppose 
this  relation  does  exist  ;  then  on  substituting  the  above  values  of 
y-ii  '  •  •  , yn  and  dividing  through  by  jj  it  becomes 

O  =  C,  +  C^f  z^dx  -j-  C^fz^dxft^dx  -\- .  •  '  -\-  Cnfz^dx  .  .  .  fw^dx ; 

differentiate  and  divide  by  z^ ,  and  we  have 

Q  +  CJt^dx  +  .  .  .  +  CJt.dx  .  .  .fw.dx  =  o  ; 

differentiate  again  and  divide  by  /, ,  and  so  on ;  we  come  obviously 
to  the  condition  d  =  o.     Retracing  our  steps  now,  and  we  find 

U„  _  I   ^^=    Un  -  2  —-   •    •     •  ^^    Cj   ^=  O. 

Denote  by  D'  the  determinant  formed  from  a  fundamental  system 
of  ^  =  o,  in  the  same  way  that  D  was  formed  from  the  fundamental 
integrals  of  P  =  o.     We  found 

(13)  D  =  C^-ZA'/.r; 
consequently 

or  D'  =  C'Dy-^, 

and 

(14)  D  =  C'D'y-. 

In  like  manner, 

<i5)  D'  =  C--D"z^-\ 

and  so  on.  Multiplying  together  these  equations  giving  D,  D' ,  etc., 
and  we  have,  finally, 


(16)  D  =  Cy,"z^''-'t^"-'  . 


w, . 


All  the  results  given  here  might  have  been  incorporated  in  the 
first    chapter,  as   they   only  require   that    the   equation   shall    have 


58 


LINEAR  DIFFERENTIAL  EQUATIONS. 


uniform  coefficients.  It  seemed,  however,  better  to  have  them  here, 
as  some  of  them  will  be  of  immediate  use.  We  will  take  up  now  the 
equation  P=  O,  as  defined  at  the  beginning  of  this  chapter,  and  see 
how  its  integrals  behave  when  we  travel  round  a  critical  point.  For 
simplicity,  let  the  point  be  the  origin,  x  -=0.  As  we  know,  the  fun- 
damental system  j, ,  .  .  .  ,  jj/„  has  the  linear  substitution  .S  imposed 
upon  it  when  we  travel  round  a  critical  point.  Suppose  5  to  be  the 
substitution  peculiar  to  the  point  x  =^o\  then  on  going  round  this, 
point  we  have 


(17) 


or 


(18) 


5  = 


^y^  =  ^.>J>',  +  •  •  •  +  c^ny„, 

Syn  =  c„^y,  +  .  .  .  +  c„„y„  ; 

^.  ;  ^n  Ji  +  •  •  •  +  c,„y„ 

f. ;  ^..j',  +  •  •  •  +  ^2«j« 

y» ;  c«ij,  +  .  .  .  +  c„„y„ 


Let  z^,  .  .  .  ,  s„  denote   another  arbitrary  system  of  independent 
integrals ;  then 


(19) 


«nj,     +    •     •     •    +   «i«^«, 


^  -«  =  a„,y,  + 


+  ««« J« . 


The  a^y  are  arbitrary  constants — such,  however,  that  the  determinant 
I  a,j  I  is  not  zero.  Suppose  that  we  travel  round  ;t-  =  o ;  then  the 
y's  submit  to  the  substitution  S,  and  consequently  the  ^''s  are 
changed  into  new  linear  functions  of  the  y's.  But  by  aid  of  the  last 
equations  we  can  express  the  ys  as  linear  functions  of  the  ^■'s,  and 
so,  after  going  round  x  ^  o,  the  2's  will  be  changed  into  linear  func- 
tions of  themselves  ;  z.e.,  they  will  have  submitted  to  a  linear  substi- 
tution. As  this  substitution  depends  in  part  upon  the  arbitrary 
quantities  a^y,  we  can  use  this  indetermination  to  affect  a  simplifica- 


LINEAR   SUBSTITUTIONS. 


59- 


tion  of  the  substitution — viz.,  we  may  seek  first  to  determine  whether 
or  not  there  exists  an  integral  which,  on  travelling  round  the  critical 
point,  changes  into  itself  multiplied  by  a  constant.  In  the  first  place, 
if  there  is  any  such  integral,  it  must  obviously  be  of  the  form 


y 


^.y.  +  •  •  •  +  «'«7«- 


Suppose  the  effect  of  going  round  the  point  is  to  change  this  into 
sy.  Apply  the  substitution  5 ;  i.e.,  travel  round  x  =  o ;  we  must 
have,  by  hypothesis, 

«^i(^u  J,  +   •    •    •   +  ^i«J«)  +  •    •    •  +   ^niCn.y,  +  .    .    .   +  C„„J„) 
=   •y(«',  Ji  +   •    •    •   +  ^«^«)- 

As  no  linear  relation  with  constant  coefificients  can  exist  among  the 
I  integrals  ji,  .  .  .  ,  y„,  this  must  be  an  identity,  and  we  have  there- 
!  fore,  on  equating  the  coefificients  of  each  y,  the  equations  of  condi- 
tion 


(20) 


r  (^n  -  •y)«',  +  ^..^.  +  •-.-{-  c„,a„  =0, 

^n^i   +   (^2.  —    -J)^.  +   •    •    •   +  ^n2  0l„  =   O, 


^i« «',  -\-  c^„a^  _j_  .  .  .  -|_  [c„„  —  s)a„    =  o  ; 

I  giving  for  the  determination  of  s  the  algebraic  equation  of  «**"  order 


(21) 


J  = 


^11  S,  c^]  ,       .    .    .    ,       C„^ 


^i«  y 


^2«  > 


J    ^tin  ^ 


=  o. 


This  equation  will  be  called  the  cJiaractcristic  equation. 

Suppose  //  =  o  has  all  of  its  roots  unequal ;  then  if  Sj  denote  one 
of  them,  and  if  this  value  of  s  be  substituted  in  the  preceding  equa- 
tion, these  will  obviously  serve  for  the  determination  of  the  ratios 
oi  a^,  a^,  .  .  .  ,  a„,  which  is  all  that  is  necessary.  The  integral  y 
will  therefore  be  known,  and  as  all  the  roots  of  A  are  unequal,  we 
will  obviously  have  n  integrals  each  possessing  the  required  property; 


•6o 


LINEAR  DIFFERENTIAL   EQUATIONS. 


or,    in    other   words,    for  this   set  of  integrals   the   substitution    5 
becomes 


(22) 


5 


y. ; 

•y.j, 

J. ; 

•y.j. 

yn\ 

Snyn 

It  can  be  shown  in  a  moment  that  these  integrals  constitute  a 
fundamental  system ;  i.e.,  are  not  connected  by  any  such  linear 
relation  as 

^1  J:    +   •    •    •   +  C„yn    —  O. 

Suppose  such  a  relation  does  exist ;  then  on  applying  the  substitu- 
tion 5,  that  is,  turning  round  the  critical  point  ^  =  O,  a  similar 
relation  must  exist,  and  so  for  any  number  of  turns  round  ;tr  =:  o. 
Suppose  n  —  i  turns  to  have  been  made ;  then  we  have 

^iJf':   +  •     •     •   +  ^nyn  =   O, 

^i^iy^  +  •  •  •  +  <^«-y«j«      =  o. 

As  the  constants  c^,  .  .  .  ,  c„  are  not  zero,  we  must  have 


I, 


I, 


=  o. 


This  is  the  product  of  the  differences  S/  —  Sy  ,  and  cannot  therefore 
vanish  unless  at  least  two  of  the  roots  s  become  equal  ;  but  by 
hypothesis  the  roots  are  all  different;  this  determinant  can  therefore 
not  vanish,  and  hence  no  such  relation  as 


^.7i  +  ♦  •  •  +  ^«J«  =  o 


LINEAR   SUBSTITUTIONS. 


6r 


can  exist ;  the  integrals  constitute  therefore  a  fundamental  system. 
If  we  had  chosen  any  other  set  of  fundamental  integrals,  say 
v^,  .  .  .  ,  Vn,  the  substitution  5  would  have  the  form 


(23) 


5  = 


^'. ;    ri 


•  •  +  Ki«^'« 


Vn  ;         r«i  ^'i    +   •     •     •   +    YnnVn 

The  characteristic  equation  is  now 


(24) 


A.  = 


Kn- 

s, 

Y-.i^ 

•  ,     Y" 

r.n' 

Y..-  s,       . 

•  ,    r« 

/  i«  >  /  2«  : 


r^^w 


The  integrals  j/^ ,  .  .  .  ,  j/,i  now  become  linear  functions  of  i\  ,  z\  ^ 
.  .  .  ,  z'n  and  reproduce  themselves,  multiplied  respectively  by  j',  ,  .  .  . , 
s„ ,  and  it  therefore  follows  that  the  equation  J,  =  o  is  identical 
with  J  =  o,  since  it  has  the  same  roots.  The  coefficients  of  z/  being 
independent  of  the  choice  of  fundamental  integrals  are  therefore  in- 
variants. A  direct  proof  of  this  fact,  viz.,  that  the  characteristic 
equation  is  the  same  whatever  set  of  fundamental  integrals  is  chosen^ 
will  now  be  given.     We  have 


(25) 


^]'^   =  Yii^\   +  •  •  •  +  r2«'^«' 


with  the  characteristic  equation 


(26) 


A.  = 


Yn    —   ^y       Y^i  ' 

Kl2  >  K22         ■^f 


ri«= 


Y2 


Yni 

K«2 


—  s 


=  o. 


62  LINEAR  DIFFERENTIAL  EQUATIONS. 

Again :  we  must  have 


(27) 


.  Vn    —    '^my,   +   .     .    .   +  A„„  jJ/„  . 


The  determinant  |  T^ij  \  is  of  course  not  zero.  Substitute  these 
values  o{  z\,  .  .  .  ,  v,^  in  the  right-hand  members  of  equations  (25), 
and  we  have 

<28)  Svi  =  (^mA,,  +  r/2A„  +  .  .  .  +  r/«A„i)j.  +  .  .  . 

.Again :  in  equations  (27)  apply  the  substitution  S,  and  we  have 
'(29)  Svi  =  {c.^Xi,  +  c,X-2  +  .  .  .  +  c„,Xi„)^^  +  .  •  . 

+  (Czn^h  +  <^2«A/2   +  .    .    .  +  c„„X,„)j/„. 

Prom  (28)  and  (29)  we  must  then  have 

k  —  n  k  =  n 

Denote  by  ^^  the  determinant  |  A,y  |  ;  we  see  then  that  the  two 
-determinants  formed  from  A  by  multiplying  it  by  z/,  and  A^ 
respectively,  have  their  elements  equal  each  to  each,  and  that  each 
of  these  products  is  equal  to 


6,,  —  A„5,      S,,  —  A,^5,      .  . 

^«i—  A„,j,     <y„2—  K2S,     .  . 
We  have  then 

A  =  A,. 


=  A. 


that  is, 


LINEAR  SUBSTITUTIONS. 


63 


This  proof  is  due  to  Hamburger.*  The  following  theorem  is  due 
to  the  same  author,  and  is  taken  from  the  same  paper: 

Theorem. — If  for  a  given  value  of  s  all  the  minors  of  order  v  in 
'//  vanisJi  {and  consequently  all  of  higher  orders),  and  those  of  order 
V—  i)  do  not  all  vanish,  theft  all  the  nmiors  of  A^  of  order  v  will 
\janish,  and  those  of  order  {v  —  \)  will  not  all  vanish. 

We  have 


A  = 


A, 


i3o)       \ 


A„  = 


^n- 

s, 

^21   » 

<^.y 

^=2-    ^, 

C^ny 

^in  » 

Vii- 

s, 

r.i  • 

r,2, 

^22—  -y* 

y^n, 

y^n. 

K, 

^xi  y 

^■,1 » 

^^,y 

^niy 


'*-«2  ? 


^  -ntf.  •* 


ym 
yn2 

ynn 


—  s 


y        ^i« 

)  "-2*1 

y       "-nn 


A,j^,      d,^  —  A„J,     .   .  .   ,   d^„—  \,„s 
X^,s,      6^^  —  A^^^,     .  .  .  ,  d^„—  X^„s 


O  fti  '^ni^y       ^  m  ^n2^y     •    •    '    y    ^  nn  '^hkS 

Let  A,-^ ,  Bik ,  Cik ,  Dik  denote  respectively  the  minors  of  order 
of  the  determinants  A,  A^,  A^,  A^\  where  i  and  k  denote  any  of  the 


nin  —  \)  .  .  .  {n  —  V  -\-  \) 
I  .2.3.  ...  V 


=  /^, 


ombinations  of  v  different  numbers  formed  out  of  the  series  i,  2, 

.  ,  71.     We  have  now 


31)    Dik  =  Q.A,,  +  .  •  .  +  C,-^Ak^  =  Bi^Ck.  +  .  .  .  +  ^/M  C, 


k\t.' 


yotimal  ftir  reine  und  angewandte  Mathemaiik,   vol.  Ixxvi.  p.  113. 


64 


LINEAR   DIFFERENTIAL   EQUATIONS. 


Since  by  hypothesis  all  of  the  minors  Ays  are  zero,  it  follows  that  all 
of  the  minors  Drs  are  also  zero  ;  we  have  then  the  system  of  equa- 
tions 


(32) 


'I   '-'21        I       •     •     •       I       -^z"ja^2;u.     ^j 


The  determinant  of  this  system,  viz., 

C  C 

C  C 

c  c 

is  a  power  of  A„,  and  therefore  docs  not  vanish  ;  consequently  we  have 
(33)  Bu  =  Bi^  =  .  ,  .  =  Bi^  =  o  ; 

and,  as  these  equations  hold  for  any  value  of  i,  it  follows  that  all 
the  minors  B^s  of  order  v  of  the  determinant  Z/,  are  equal  to  zero. 
If  now  all  of  the  minors  of  order  {v  —  i)  of  z/,  vanish,  it  is  clear  from 
the  foregoing  that  all  the  minors  of  J  must  also  vanish,  which  is  con- 
trary to  the  hypothesis. 

We  will  take  up  now  the  case  where  the  characteristic  equation 
has  equal  roots. 

By  employing  the  results,  not  yet  mentioned,  of  Hamburger's 
paper  the  forms  of  the  integrals  can  be  obtained  very  readily ;  but 
it  seems  better  to  first  obtain  these  forms  in  the  same  way  that 
Fuchs  obtained  them,  if  for  no  other  reason  than  that  of  the  de- 
sirability of  developing  the  subject  in  historical  order. 

From  what  precedes  it  is  clear  that  there  is  always  at  least  oneB 
integral,  say  ?/;,  which  is  changed  into  s^u^  when  the  independent^ 
variable   travels  round    the  critical  point  ;tr  =  o.     l(  f^,  .  .  .  ,  V» 
denote  a  system  of  fundamental  integrals,  we  have 


(34) 


>'iJi  +  •  •  •  +  «« J« . 


PROPERTIES  OF   THE  INTEGRALS. 


65 


where  the  coefficients  a^,  .  .  .  ,  a^  are  not  all  zero.  It  is  clear  now 
that  we  can  replace  one  of  the  integrals  j, ,  .  .  .  ,  yn  by  u^  and 
still  have  a  fundamental  system,  provided  that  the  coefficient  a 
corresponding  to  the  replaced  y  is  not  zero.  Suppose  a^  different 
from  zero  ;  then  we  can  take  for  a  new  fundamental  system  the 
integrals 

u^,  y.,  y.^   •  •  •  .  yn- 

The  result  of  going  round  the  critical  point  ;r  =  o  is  now  to  change 
these  integrals  into  the  following  : 


(35) 


^^y.  =  /3„jf,    +  A,j,  +  .  .  .  +  /3,„y„ , 

^y.  =  A.?^  +  A.J.  +  •  •  •  +  A«/« ' 


^  Sy„  =/?„,?/,  +  /3„^y^  +  .  .   .  -\-  /3„„y„. 

The    characteristic    equation    corresponding  to    this  system   is  ob- 
viously 

/3,,  ~  s,     /3,,,  .  .   .   ,  /?„, 

A3 ,  /?33  -  J-,     .   .  .   .  /i„3^ 


(36) 


{s,  -  s) 


ft.n, 


Pj,n  1 


P  nn 


o. 


As  jj  is  a  multiple  root  of  the  characteristic  equation  z/  =  o,  and  as 
the  roots  of  the  characteristic  equation  are  quite  independent  of  the 
choice  of  a  fundamental  system  of  integrals,  it  follows  that  the 
equation 

Pii  ^J  •      •      •      >  Pfl2 

(37)  :  :  =  o. 


P2n  y 


/3„n   —    S 


has  at  least  one  root  s  =  s^.     This  being  so,  it  follows  further  that 
there  must  exist  a  compatible  system  of  equations,  such  as 


(38) 


A,^,         +  (As-  s,)A,  + 


=  o, 
=  o, 


HttiA^ 


+   As^s 


+ 


+  (/5««  -  s,)An  =  o. 


66  LINEAR  DIFFERENTIAL  EQUATIONS. 

If  now  we  write 

«,  =  A^y^  +  A,y,  -|-  .  .  .  +  Any„ , 

and  then  travel  round  the  critical  point  ;r  =  o,  we  shall  have 

Szi,  =  {^,,A,  +  /3,,A,  +  .  .  .  +  /3„,A„)u,  +  s,u, , 
or 

(39)  S^-,  —  s,,u^  +  s,u, , 

where  s„^  is  a  constant.  The  new  function  u^  is  obviously  an  inte- 
gral, and  we  can  replace  by  it  any  one  of  the  integrals  jj/, ,  .  .  .  ,  y„ 
of  which  the  coefificient  a  is  different  from  zero.  Suppose  that  «, 
replaces  _;j/.j  ;  then  we  have  a  new  fundamental  system, 

^i,     u^,    jj/3,     .  .  .  ,    y„. 

If  J,  is  a  triple  root  of  the  characteristic  equation,  we  can  find  a  new 
integral  ii^  satisfying  the  equation 

where  ^3,  and  s^^  are  constants.  If  then  s^  is  root  of  the  characteristic 
equation  of  multiplicity  A,  we  can  obviously  form  a  group  of  X  inte- 
grals ?i^,  zi^,  .  .  .  ,  7ix  ,  such  that 

5//,     =  5-,//, , 
41)  is?/,    =  s,j(,   -{-  s^^u,  -\-  s,t/,, 

Finally,  if  the  distinct  roots  of  the  characteristic  equation  J  =  o  J 
are  .jj ,  s^,  .  .  .  ,  S/ ,  and  if  their  orders  of  multiplicity  are  A,  ,  A,, 
-  .  .  ,  A.^,  we  can  form  in  the  manner  above  indicated  /groups  of 
integrals  containing  in  all  A,  -[-  ^^2  +  •  •  •  -\-  ^i  =  '^  integrals,  and 
these  integrals  will,  from  what  precedes,  form  a  fundamental  system. 
By  aid  of  the  above-found  properties  of  a  system  of  fundamental 
integrals  it  is  easy  to  find  the  forms  of  the  integrals. 


FORMS  OF   THE  INTEGRALS.  67 

Write 

log  5. 


■(42)  r,  = 


2Ttl 


log  j'l  standing  for  any  one  of  its  values.  Suppose  u^  an  integral 
such  that  when  the  variable  turns  round  the  critical  point  x  =^  o  we 
have 

<43)  Sii,  =  s^u^ ; 

then  it  is  clear  that  in  the  region  of  this  point  the  product  x~*'^ii^  is 
a  uniform  function,  say  0,(-^)  >  then 

(44)  ti,  =  X-'  0,(a-). 

It  follows  at  once,  if  all  the  roots  of  the  characteristic  equation  are 
•distinct,  and  if 

(45)  '•.=  '-"If*, 
that  each  integral  can  be  written  in  the  form 
■(46)                                             Uk  =  x^'k  (pkix)  ; 

where  <Pk{x)  is  a  uniform  function  of  x,  and  where  the  exponents  r, , 
r, ,  .  .  .  ,  r„  do  not  differ  from  each  other  by  integers.  This  last  is 
■clear  since 

if  r/  =  r^  -f-  in,  where  ;«  is  an  integer,  then  we  should  have  Sk  =^  si, 
which  is  contrary  to  hypothesis.  The  functions  (p{x)  are  develop- 
able in  series  containing  both  positive  and  negative  powers  of  x, 
and  convergent  in  the  region  of  ;r  =  o. 

In  the  case  where  s^  is  a  multiple  root  of,  say,  order  A,  the  ana- 
lytical forms  of  the  corresponding  integrals  may  be  found  as  follows: 
We  have  already  shown  that  there  exists  a  group  of  integrals  ti^ ,  u,, 
-  .  .  ,  ?^A  having  the  properties  expressed  in  equations  (41),  and  that 
n,  is  of  the  form 

u,  =  x'-^cb.,  ; 


68  LINEAR   DIFFERENTIAL   EQUATIONS. 

where  0ji  is  uniform  in  the  region  of  the  point  ^  =  o,  and  where 


r.  r= :  log  s.  . 


In  respect  to  zi..^,  equations  (41)  show  that 
Su 


Hence 


S?i,  \u. 


s.  II, 


Hence  the  substitution  S,  that  is,  the  result  of  a  circuit  of  the  vari- 
able  around  the  point  x  ■=  o,  reproduces  —  increased  by  a  constant ; 
the  point  ;ir  =  o  is  therefore  a  logarithmic  critical  point  for  the  func- 


tion   — ,  and  consequently 


-.  log  X 


is  uniform  in  the  region  of  the  point  ^  =  o.     Designating  this  func- 
tion for  the  moment  byy"(,r),  we  have 


—  — -■ — •  log  ;tr  =  /  {x), 


or 


^^.  =  uJ{x)-^-^^u^ogx; 


that  is,  since  u^fix)  =  ;r''i0„(;f)/(^),  where  <t>^^{x)f{x)  is  necessarily 
a  uniform  function, 


x''^ 


0n(^)/W  +  ^;^-  0n   log  X 


Making  (p^,{x)f{x)  =  (p^,{x),  we  have 


(47) 


u„  =  x''^ 


0.:  +  2^777'  '^"  '°^  ^ 


FORMS  OF   THE  INTEGRALS.  69 

In  the  same  way, 

or,  substituting  for  —  its  value  found  above, 

s{-'\  =  -'i  +  -f^^.  log;r  +--=  +/(^). 

jNow  5(log^;ir)  =  log';ir  -|-  A,7ti  log;r  +(27r/)',  and  remembering  the 
iform  of  y(;ir),  it  is  clear  that  — ^  behaves  Hke  the  function 

c  log'  X  +  i\){x)  log  ^  +  xix) ; 

where  i:  is  a  constant,  and  ip{x)  and  ;^;(;r)  are  uniform  functions.     It 
is  therefore  plain  that  we  must  take 

11 
~  =  X{^)  +  4i^)  log  A'  +  c  log'x; 

or,  giving  u^  its  above-found  value, 

(48)  ?/3  =  x'-r  103,  +  03,  log  X  -\-  033  log'  x\; 

where  03,,  03,,  and  033  are  all  uniform  functions,  and,  in  particular, 

033  =  cx-  "-m^ . 

The  law  of  formation  of  the  successive  integrals  of  this  group  is 
now  evident.     It  may  be  enunciated  as  follows : 

If  s^  is  a  root  of  the  characteristic  equation,  of  multiplicity  A, 
then,  corresponding  to  this  root,  there  exists  a  group  of  A  integrals, 

^1 J     ^2  >     •  •  •  »     U\  , 

having  the  properties  set  forth  in  equations  (41),  and  they  can  be 
put  into  the  following  form,  viz.. 


70 


LINEAR  DIFFERENTIAL   EQUATIONS. 


z/,  =  ;»:'->[  0,.  4-  0,,  log  ^], 

(49)  1  u,  =  ;tr'-i[03.  +  03,  log  X  +  033  log"  x], 

?^;,  =  ;ir''i[0A,  +  0A2  log  j:  +  .  .  .  +  0aa  log^  -  'x]  ; 
where  r.  =      ^  .',  and  0,, ,  0,, ,  .  .  .  ,  0aa  are  uniform  in  the  neigh- 

27tt 

borhood  of  the  point  x  =  o. 

(I)  The  quantities  0  are  such  that  any  one  of  them,  0^,  where  i  is 
different  from  /,  can  be  expressed  linearly  in  terms  of  those  whose, 
second  subscript  is  i. 

(11)0,,,  023,  ...  ,  0AA  differ  from  one  another  only  by  constant 
factors. 

Assuming  the  above  statements  to  hold  good  for  the  first  k  —  i 
of  the  integrals  n^ ,  21^ ,  .  .  .  ,  u^  ,  they  may  be  shown  to  hold  good! 
for  the  k^^  integral,  ?a  . 

For,  by  equation  (41), 

(50)  S?lji.   =   Sk,1l^  -\-  SkS^  +  .    .    .  ^  Sk,k-^Uk-,-\-  S.Uk. 

Also,  we  may  always  take 

(51)  Uk  =  x'-^ { <t>k.  +  0/C-2  log  ^  +  .  .  .-\-(pkk  log^- -'x\; 

<t>ki ,  0/5-3 ,  .  .  .  ,  <Pu-  being  chosen  at  will,  provided  that  0^,  is  prop- 
erly determined. 

Assume  then  that  0^^,  0^3,  .  .  .  ,  (p^/t  are  uniform  in  the  region 
of  the  point  x  =  o,  and  let  0^,  become  (p'j^,  when  the  variable  moves 
round  the  critical  point.      Equation  (51)  thus  changes  to 


Suik  =  s.x'-i 1 0'^,  +  0^3 (log  x-{-27rt)  +  .  .  .  -\-  0^^(log  X  +  2;rz)*  -  ' | .. 
Expanding  and  dividing  both  members  by  s^x''^,  we  have 

SjX''^   ~ 


I 


FORMS  OF   THE  INTEGRALS. 


+  <t>k,  k- 


k  —    I  .  k  —  2  .k  —  X 
+  (2^0 J] -<t>kk 

.     ,        ...  k  —  2  .  k  —  I 

+  (2;rz  Y jj ^0^, ,  _ , 

+  27ti{k  —  3)0^,^  _^ 


-\-27ti{k—2)<l)ji^jt_, 

log-*  -  ^X 


71 

log*-3;»r 


+  •   .  .  +  {2niY 


k  —   \  .  k  —  2  .k  —   X 

4 £ 


3! 


0, 


'kk 


,     ,       ...        -^—  2./^  —  3./^  —  4 

+  (2^0*-' ^ TT^ -Cpk.k-. 


+  (2;r?)^ 


k  —  A.k—   ^.k  —  6 


(pk,k-: 


I  .  k  —  2 


+ 

+  (27r/)^  -  3  ' ~ cp,, 

.    .      ...       k  —  2  .k  —  -x 

+  (2^0^  - ' Y\ "^'^ '  - ' 

I      /         -xi  k   —    X  .k  --  A. 

+  (27r0*  -  5 ^j -<pk,  k  - . 

I    /      -xi    A  '^  —  4  •  ^  —  5 
+  (2^0^-^ Vl ^0-^,^-3 


log';ir 


0^,4  I 


4-       . 

+  (2«) 

+ 


30.^4 

0>fe,3 


^2 


LINEAR  DIFFERENTIAL   EQUATIONS. 
+  {27liY  -  \k—  \)<t)kk  log  X  +  {27tiy  -  'Cpkk 

+   {27tiY-\k-l)cPk,k-^ 

+  {2ni)^~^{k—^)(})k,k-.^ 
+         .         .         .         . 
+         ,         .         .         . 

+ 


30^4 

20^3 
0.«-2 


^{2niY- 

'^<Pk,k-r 

+  {27Tiy- 

-    ^(pk,  k  -  2 

+  {27tiy- 

-^0^,^-3 

+      . 

• 

+      . 

• 

+  (2^0' 

0/^4 

+  (2^0' 

ft>k^ 

+  27rz 

(Pk2 

+ 

<t>'kx' 

But  by  substituting  from  equation  (50)  in  equation  (51)  we  have 


S,(pkk^Og^--'X-\-  S,(pk,k-i 


4"  S,(pk,k-z 

-{-  Sk,k  -X  0^-i,/t-3 

-|-  -^/Er,  ^-2  0/5'-2, /&-3 

+  ^/{r,  .i-3  0-4-3,/5r_3 

+  -^,0^4 

-f-  -J/t,  /{r-  I  0/^  -  1,4 

-)-  -^/t,  /t  -  2  0^  -  2,  4 

+  •^'t,  ^  -  3  0^^  -  3,  4 
+  •  •  • 

+  -^^4  044 


^X-\-S,Cf)k,k-^ 
-{-  Sk,k-l<Pk- 


k—i 


\0^k  -  4^   _|_ 


log*  -  ^X 


log*   X  +  ^,0^3 

4"  -^/i,  ^  -  I  0^  -  I,  3 
-f-  -y/t,  /t  -  2  0-J  -  2,  3 
+  -y^,  A  -  3  0/t  -  3,  3 
+  .  . 

+  Sk^  043 
+  ^k^  033 


log'^ 


FORMS  OF   THE  INTEGRALS. 


n 


+  •^10^2 

+  •$■/&,  /fe  -  3  0/i  -  3,  2 
+  •  •  • 

+  ^k^  042 
+  Sk^  032 


log  X  -\-  S,(f)k^ 

+  ^i,k  -  z4>k  -  i,z 

-\-  Sk,k-2<Pk  -   2,1 

+  -y/t,  /&  -  3  0;i  -  3,  1 

+  •     •     . 

+  -^.^4  041 
+  Ski  031 

+  ^kz  021 

+  Sk^  011 


"VVe  thus  find  two  distinct   expressions  for 


Suk   . 


x^^ 


in  terms  of  posi- 


tive integral  powers  of  log  x.  Identifying  the  coefficients  of  the 
same  powers  of  log  x  in  the  two  expressions,  the  following  equations 
are  obtained : 


I 


(52) 


•^10^   —   S^(pkk  , 

S\27ci{k  —    \)(t>kk-\-  (t>k,k  -i'\   —   S,(pk,k-T.-\-  Sk,k  -■L<Pk--i,k-x, 

i  k—  I  .  k—2  ) 

S,  \    {27tiY ^^ -(pkk  +  {27ti){k—2)(pi^  k-i-\-  (pk,k-2    Y 


^l^Pk,  k-2-\-  Sk,k   -    l4'k  -    l,k    -   2-\-    ^k,    k   - 


2H'k  —  2,    k  —  z 


The  whole  number  of  these  equations  is  k,  but  the  first  is  only  an 
identity.     Observing  that  the  [k  —  i)^'^  equation  is  of  the  form 

.y,|  .  .  .  +  2(27rz)0^3  +  0^2}  =  ^10,^2  -^  Sk,k-x(t>k-i,2-^  '  •  .  , 

'  the  terms  in  0^2  are  seen  to  vanish  identically ;  hence  the  2d,  3d, 
.  .  ,  ik—  i)^*  equations,  in  number  k—2,  involve  only  the  k  —  2 
I  unknown  quantities 

<Pkk  ,        <pJi,i-i,        •    •    '    ,        0^3  » 


which  can   therefore   be   expressed   in  terms  of  0's  of  lower  index 
supposed  already  known. 


74  LINEAR  DIFFERENTIAL   EQUATIONS. 

The  determinant  of  the  left-hand  members  is 


2Tti(k—\), 

k—\.k—2. 


{iTiif  ■ 


o, 


(2ni){k  —  2.,)  o. 


^„k—\.k  —  '2.k—'\     ,        ^^k  —  l.k—'X 

{iTiif— j ,  {2nif j ,    2;r/(/6— 3),   o, 

3  •  2  . 


.^,      k—^.h—l-k—"},     ,      .,.      k—^.k—i.k—^ 

{2nif-^ ; -,     {21nf-'= ~ 


{litif-'i 


k—\.k—1 


{2TtiY-^ 


k-2.k  —  2 


{2Ttif-\k-i), 


(2ff/)*-3(/J  — 2), 


o,         o 
o,         o 

o,         o 


•  .  o,         o 

.  ,  3(27rz),         o 

.   ,  2i{2nif,  2{27t{) 


of  which  the  value  is 

s,^-^k-\  .  k^'-2  .  .  .3.2.  {27tiy  -  ■"  =  {27ii)^  -  ''s,^  -  ■^ .  {k—  i) 
Replacing  the  columns  of  this  determinant  successively  by 

Sk,k  -i4'k  -  r,  k  -    -ii 

^k,k  -\4^k  -  1,   k  -  2  ~r  -^-f,  >5r  -  2  0/{r  -  2,  k  -  ■zy 

Sk^k  -  i(pk  -  T,i  -\-  Sk,k -^<Pk-^,2-\-   '    '    '   -\-  ^i,  2022  » 


and  dividing  the  results  by  (2711) 


k  —  2  r   k  —  2 


{k  —  i) ! ,  the  values  of 


cpkk,  •  •  •  ■,  <Pk,i  are  obtained  in  order.     In  particular  it  is  found  that 


(53) 


<Pkk  = 


27ii{k  —  \)s^     * 


which  establishes  proposition  (II),  since  k  may  have  any  value  from 

2  to  A.     Also, 


FORMS  OF   THE  INTEGRALS. 


n 


(54) 


03!,    = 


2, 


— r\0s 


S,{27ti) 
2711,       J,,0„   +  -^31011 

•y.,0u 


-i-  S,{27ti)  .  2!  . 


But  it  is  already  known  that  0„„  =  -T^-^^x '  hence  0,,  is  expressed 

t  in  terms  of  0„  ,  0,^ .     Proceeding  in  the  same  manner,  it  may  be 
seen  that  proposition  (I)  holds  good  throughout. 
We  have  also  the  equation 

sM^^^y  -  '0**  +  (2^0*  '  V.,*  -  ,+  ...  +  (27rzy0.4  +  (2^0>*3 

+  (2;r/)0i,  +  0A,| 

=    J,0i,    +   -y*.  *  -  I0A  -  I,  I  +    .    .    .    +   ^*404i    +   •^*303i    +  •^A202.    +   Ski<Pii  - 

ct)k2  is  by  hypothesis  an  arbitrary  uniform  function;  it  may  therefore 
be  so  chosen  as  to  satisfy  the  following  equation  : 

s,\{^niY-'(pkk  +  .  .  .  +  (2;rz)'0i.3  +  {'2^i)(pk.\ 

=   -y*,  A  -  I0A  -  I.  I  +  •   •   •   +  •y*202i   +  •^41011  •• 

Hence  we  must  have 

(55)  -^10*1  =  -^10*1; 

that  is,  4>ki  is  also  a  uniform  function  in  the  region  of  the  point 
X  =  o. 

It  is  thus  shown  that  if  the  law  of  formation  of  the  integrals  u^  ,, 
11^,  .  .  .  ,  j(x,  as  already  stated,  holds  good  for  the  first  k—  i  oi  them, 
it  also  holds  good  for  the  next  following.  But  it  was  shown  by 
direct  investigation  to  hold  for  11^  and  u^\  it  is  therefore  true  for  the 
whole  group. 

From  the  equation 


0A 


^k.  k 


we  find  at  once  that 
(56)  <t>kk  = 


2ni{k  —  \)s^ 


i,k- 


{k  —  \){k  —  2)  .  .  .  I  .  .y,*  -  '(27rzy 


1011- 


76  LINEAR  DIFFERENTIAL  EQUATIONS. 

In  the  group  of  integrals  thus  obtained,  any  one  of  them,  21^1  may  be 
replaced  by  a  linear  function  of  the  integrals  of  lower  index  without 
altering  the  form  it  assumes  after  undergoing  the  substitution  5 ; 
since  the  integrals  involved  will  all  be  reproduced  increased  by  linear 
functions  of  those  of  lower  index  only.* 

The  system  of  integrals  (49)  satisfying  the  conditions  (41)  and 
corresponding  to  the  root  s^  of  order  of  multiplicity  A  is  called  by 
Fuchs  ^  group.  We  have  found  that  the  uniform  functions  0/;-  can 
all  be  expressed  linearly  in  terms  of  those  whose  second  sufifix  is  i, 
and  in  particular  that  the  functions  0,, ,  02, ,  •  .  •  ,  0aa  differ  only 
by  constant  factors. 

Corresponding  now  to  any  linear  differential  equation  with  uni- 
form coefficients,  we  have  a  certain  number  of  distinct  groups  of 
integrals  corresponding  to  each  critical  point.  Corresponding  to  the 
point  ;tr  =  o,  suppose  the  characteristic  equation  has  k  distinct 
roots  i'l ,  ^2 ,  .  .  .  ,  Sk,  o{  orders  of  multiplicity  A, ,  A^,  .  .  .  ,  A^.,  re- 
spectively ;  then  Aj  -(-  A^  -f~  •  •  •  +  >^/fe  =  ^^>  and  there  are  k  groups 
of  integrals  of  the  form  given  in  equations  (49).  By  aid  of  the 
theorem  proved  above  concerning  the  vanishing  of  the  minors  of  A 
and  zJ, ,  Hamburger  has  shown  how  each  of  these  groups  gives  rise 
to  certain  sets  of  sub-groups  which  are  independent  of  one  another, 
and  such  that  the  linear  relations  connecting  the  uniform  functions 
above  denoted  by  0,y  {i.e.,  the  coefficients  of  the  different  powers 
of  the  logarithm)  exist  for  each  sub-group  separately.  In  other 
words,  there  is  no  relation  connecting  any  of  the  uniform  functions 
of  one  sub-group  with  those  of  another  sub-group.  Hamburger's 
work  (Crelle,  vol.  ^6)  is  an  extension  of  a  process  due  to  Jordan 
{Comptcs  Rcndus,  t.  Ixxiii).  Floquet  {Annales  d'Ecole  Norniale 
Supe'ricure,  2'  serie,  t.  12)  has  also  used  Hamburger's  results  in  con- 
nection with  linear  differential  equations  with  periodic  coefficients. 
Hamburger's  determination  of  the  sub-groups  of  integrals  which 
immediately  follows  is  taken  with  but  very  slight  alterations  from 
his  paper. 

We  have  j)', ,  .  .  .  ,  j„  for  a  fundamental  system  of  integrals,  and 
for  any  other  integral 

(57)  u  =  a,y,  +  .  .  .   -f  a„y„ ; 

*  The  preceding  verification  of  (I)  and  (II)  was  worked  out  by  Mr.  C.  H.  Chapman. 


SUB-GROUPS  OF  INTEGRALS. 


77 


for  the  integrals  under  consideration  we  have  S21  =  s^ti,  and  for  the 
equations  determining  the  ratios  oi  a^,  .  .  .  ,  (x„, 


158) 


(^n  —  -^i)  ^i  +  ^2i«'2 + 
^..«'i  +  (^"22  —  -^i)  ^2  + 


+  (^,na„  =  O, 


^i>i^\      I     ^2?:  "2 


+     .     .     .     -f  (C„„  —  i-J  O',, 


If  we  had  chosen  z', ,  .  .  .  ,  7'„  as  our  set  of  fundamental  integrals, 
we  should  have  (equations  23  and  24) 


(59^ 


(r.i  — -^i)  «^i  +  r=.«'2  +  •  •  •  +r«i«^«  =0' 

r,.^'.  +  (r22  —  -y.)    +  •  •  •  +  r«2«^«  =  o* 


Ki;/^,  -f-  K2««'2 


+     .     .     .     +  {ynn  —  S,)  «'„  =  O. 


The  characteristic  equations  z/  =  o  belonging  to  the  system 
./,,...,  j'n ,  and  z/j  =  o  belonging  to  the  system  t\  ,  .  .  .  ,  v,^ ,  have 
the  same  roots.  Now  from  what  has  been  shown  above  (equations 
33  ct  seq.)  it  is  easy  to  see  that  there  are  the  same  number  of  inde- 
pendent equations  in  the  system  (59)  as  in  the  system  (58).  For,, 
the  necessary  and  sufficient  condition  that  n  —  v  -\-  i,  and  no  more, 
equations  of  either  system  are  dependent  upon  the  remaining 
equations  of  the  system  is  that  all  the  minors  of  order  v  of  the 
corresponding  determinant,  J  or  zJ, ,  shall  vanish,  while  all  those 
of  order  v  —  i  shall  not  vanish — a  condition  which,  as  we  have 
proved,  is  simultaneously  satisfied  by  both  J  and  //, .  Since  s,  is  a 
root  of  J  =  o,  it  follows  that  at  least  one  of  the  system  of  equations 
(58)  is  a  consequence  of  the  remaining  ones.  There  may,  however,, 
be  more  than  one  of  the  equations  (58)  which  are  dependent  upon 
the  remaining  equations  of  the  system.  Suppose,  for  example,  that 
there  are  n  —  v  independent  equations.  Now,  since  the  choice  of  the 
system  of  fundamental  integrals  does  not  at  all  affect  the  character- 
istic equation,  it  follows  that  the  number  of  independent  equations 
in  (58)  and  (59)  is  the  same;  that  is,  the  number  of  independent 
equations  is  the  same  whatever  set  of  fundamental  integrals  is  em- 
ployed. Assuming  now  n  —  v  independent  equations  in  the  system 
(58),  we  can  determine  n  —  v  oi  the  constants  a^,  .  .  .  ,  a„  in  terms 


78 


LINEAR  DIFFERENTIAL  EQUATIONS. 


of  the  remaining  ones,  or,  what  is  the  same  thing,  we  can  determine 
all  of  the  constants  <a'i ,  .  .  .  ,  o'„  as  linear  homogeneous  functions* 
of  V  arbitrary  constants.  It  follows  then  that  all  of  the  integrals 
which  satisfy  the  relation  Sii  =  s,u  are  linearly  expressible  in  terms 
of  V  linearly  independent  functions  which  we  will  denote  by  ?/,, 
n^,  .  .  .  ,  ?/„.  We  may  replace  now  v  of  the  fundamental  integrals 
jVj ,  .  .  .  ,}'„hyu^,  .  .  .  ,  u^ — say  the  new  fundamental  system  is 

u,,  u,,  .  .  .  ,  z/^./^  +  i,  .  .  .  ,j„; 
and  then 

(60)     5  = 

«1  ,    It-i  ,  .  .  .  ,   Uv  ;  SiUi  ,  Ji«j  ,   .    .   .    ,  SiUv 

;V  +  i  ;<:  v+i,  I  Ml  +  ...  +  ^V+i,  V  Uv  -\-c'v-{.i^  v-{-i  yv-\-\-\-  .  •  •  +  <^V-1-:,  nyn 

Vn  ■,c'nitl\         -\- ...-{■  c  nv  Uv  -f  f '«,  iz+i  j)V+i    -\-  .  .  .  -\- c' nn yn 

The  characteristic  equation  becomes  now 


(61) 
where 


A  ={s  —  j-j"  A'  —  0\ 


(62) 


A'  = 


^    v-\-i,v-\-i  •^»     •     •     •     )^«,  »'-)-I 


^  v-\-  1,  n  > 


^    nn  ^ 


As  the  characteristic  equation  is  independent  of  the  choice  of  the 
fundamental  system  of  integrals,  it  follows  that  equation  (61)  has 
the  same  roots  as  the  original  equation  J  =  o ;  and  if  s^  is  a  root  of 

multiplicity  A,  we  must  have  v  _  X.     li  v  =  X,  then  u^,  .  .  ,  u^  are 

all  the   integrals  associated  with  the  root  s^  and  satisfying  the   rela- 
tion Sii  =  s^u.     We  have  in  this  case  v,  =  A,  sub-groups  of  integrals 

*  As  we  have  principally  to  do  with  linear  homogeneous  functions,  it  will  be  con- 
venient to  drop  the  word  "  homogeneous,"  so  that  a  "  linear  function"  of  any  set  of 
quantities  will  be  understood  to  mean  a  "  linear  homogeneous  function"  of  those 
quantities. 


SUB-GROUPS  OF  INTEGRALS.  79 

each  containing  one  member.  If,  however,  v  <A,  then  obviously 
J,  is  again  a  root  of  the  equation  A'  =  o.  This  equation  implies  the 
existence  of  a  system  of  linear  equations  of  the  form 


(63) 


'  a'/(<:^  +  i,  ^  +  x  — -y,)  +   •   .  •   +  «'«-„^'„,  „  +  i        =  O, 

«',Vv+,,  „  +    •     •     •     -\r  ^'n-y{c'„n  —  S^)  =  0, 


which  serve  to  determine  the  ratios  of  the  constants  a/,  .  .  .  ,  a'„  _  y. 
If  we  multiply  the  ;/  —  v  equations  corresponding  to  the  last  n  —  v 
lines  of  the  substitution  (60),  we  have,  by  aid  of  (63),  for  the  new 
integral  sought  the  equation 

(64)  V  =  a/jy  +  .  +  .  .  .  +  «'«  -  uf,, , 

and  this  from  what  precedes  is  easily  seen  to  satisfy  the  relation 

(65)  Sv  =  s,v  +  U\ 

where  f/ is  a  linear  function  of  u^,t{^,  .  .  .  ,  ii^.  If  now  we  find 
that  there  are  only  n  —  v  —  v'  independent  equations  in  the  system 
{63),  then  it  follows  that  we  can  determine  the  n  —  v  constants 
«'/,...,  a'„  _ ;,  as  linear  functions  of  v'  perfectly  arbitrary  con- 
stants, and  therefore  there  will  exist  v'  linearly  independent  integrals 
satisfying  the  relation  (65). 

Say  these  integrals  are  i\,  v^,  ,    .  .  ,  v^,> ;  then 

{66)  Sz\  =  s,z\  +  U, ,    Sv^  =  s,v,  +  U,,    .  .  .  ,    Svy'  =  s,v,>  +  C/,' . 

where  l/^,  [/,,...,  Wy'  are  linear  functions  of  i/^ ,  u^,  .  .  .  ,  7^^. 
Every  other  integral  satisfying  the  relation  (65)  will  be  a  linear 
function  of 

M,,       tl^,       .    .    .    ,       tlv,       V^,       V^,       .    .    .    ,       V^,. 

Further,  the  functions  U^,  U^,  .  .  .  ,Uv'  are  linearly  independent ; 
for,  if  they  were  not,  it  would  be  possible  to  form  a  linear  function 


8o  LINEAR  XiIFFERENTIAL   EQUATIONS. 

oi  i\,  v^,  .  .  .  ,  Vv> ,  say  V,  which  would  satisfy  the  relation  SV=  s^V;. 
that  is,  there  would  be  more  than  v  linearly  independent  integrals, 
satisfying  this  relation,  which  is  contrary  to  hypothesis.  It  follows, 
also  that  v'  cannot  be  greater  than  v,  since  between  any  r  -\-  i  oi  the 
functions  Ui  there  necessarily  exists  a  linear  relation.  Finally, 
since  Uj  is  a  linear  function  of  ?^j ,  .  .  .  ,  u^,,  and  since  each  of  these 
functions  satisfies  the  relations  Sui^=  sjii,  we  have  SUi^^  s^Ui,  and. 
writing 

Z7=C,^,+  .  .  .  C.'£4', 

where  d ,  •  •  •  ,  Ci''  are  constants, 

If  now  V  -\-  v'  =:  \^  then  the  A  integrals  corresponding  to  the  root  s^ 
of  the  characteristic  equation  are  divided  into  v'  sub-groups  of  two 
elements  each,  and  v  —  v'  sub-groups  of  one  element  each  ;  viz.,  the 
sub-groups  of  two  elements  are 


(67) 


r  {v, ,     U^)  satisfying  S7\  =  s,v^  +  U^ ,  SU,  =  s^U,, 


(7v ,  f/v')     •     .     .     -^'V'  =  s^v^'  -\-  U^' ,  SU^'  =  s^  Uy' ; 


and  the  sub-groups  of  one  element  each  are  the  remaining  v  —  v' 
linear  functions  of  11^,71^,  .  .  .  ,  ?/^,  which  have  no  linear  relation 
among  themselves  or  with  [/^ ,  .  .  .  ,  Uw .  There  will  obviously  be 
no  loss  of  generality  if  we  replace  U, ,  .  .  .  ,  U^f  simply  by  zi^  ,  .  .  .  , 
?/^',  and  the  remaining  T' —  i^' functions  by  ?v  +  i ,  •  •  •  ,  2^^ ;  denoting 
then  by  II  the  sub-groups  of  two  elements  each,  and  by  I  the  sub- 
groups of  one  element  each,  we  have 

(68^  (  II  =  (?^  .  ^^.)'     (^^='  ^0.     •  •  •  '     i^^"' '  M> 

where  Sv^  =  s^Vi  -\-  ?/, ,     Sui  =  sji,- . 

Suppose  now  that  v  -\-  v'  <\;  then  we  can  choose  for  our  fun- 
damental system 

U^,    .    .    .     ,    lly,        7',  ,     .    .     .     ,    TV,       y^  +  v'+iy    •    '    •    ,   yn', 


SUB-GROUPS    OF  INTEGRALS. 


for  which 

(69)     5  = 

H\  ,   .  .  .   ,1lv  \  Si^i  ,    .  .  .  ,  SiUv 

7'i  ,    .  .  .    ,   7'v'  ;   SiT'i  -\-tti SiVv'  -\-  Uv' 

yv-\-v'-\-i  ;  c"v-\-v'-{-iU\-\-  ...  -\-c"v-'rv'-\-i,v»v-\-c"v-\-v'-\-i,v+iVi-{-  ...  -\- c"  vA-v' A- T ,  n  V  n 


yn\  c'nxU-i,  -\-  .  .  .-\-c"  nvtlv-\-c"  n,v-^-iV\ 

The  characteristic  equation  is  now 

(70)  A^{s-  s^^-^^'  A"  =  0; 

where 


-|-  .  .  .  -\-c",t,iyn  I 


." 


(71)    //"  = 


^    V  -{-  v'  -\-  '',  V  -\-  v'  -{- 1  J  >     •     •     '    f   f-     «,  r  -|-  v'  -|-  I 


+  I''  4-  I,  « 


Since  v  -{-  r'  <  A,  it  follows  that  5,  is  again  a  root  of  J"  =  o,  and 
this  equation  implies  the  existence  of  the  system 


^1"  {^""  +  "'+1,  v  +  v'+i  ^i)~\~   '    •    '    ~{'^     n  -  V  -  v'C" n^  I'  +  ^'  +  i  =  O, 


(72) 


-v-\-v' -\-\,  n 


+   .    .    .  -\rOc"n_^_^\c"„n—  S^  =  0, 


serving  to  determine  the  ratios  of  a-/',  .  .   .   ,a'"„_^_^'.      Multiply- 
ing the  n  —  V  —  v'  equations  corresponding  to  the  last  ;/  —  v  —  v' 
\       rows  of  (69)  by  «/' ,  •  •  • ,  a"«  _v_v',  and  adding,  we  have  for  the  new 


function 


^  =  ^'/'j^.'+f'  +  i  +   •    •   •    +«"«-^-..'JF«, 


which  is  linearly  independent  of  the  tis  and  the  ^-'s,  and  which  satis- 
fies the  relation 


(74) 


Szv  =  s^7v-\-  T; 


where  T  is  a  linear  function  o(  u^,  .  .  .  ,  ti^ ,  7\,  .  .  .  ,  Vy> .    Suppose 
now  that  there  are  only  11  —  v  —  v'  —  v"  independent  equations  in 


82 


LINEAR  DIFFERENTIAL  EQUATIONS. 


the  system  (72).  Then  all  of  the  n  —  v  —  v'  constants,  a^' ,  ,  .  .  , 
a" n^v-v' ,  are  linearly  expressible  in  terms  of  v"  arbitrary  constants, 
and  consequently  there  exist  v"  linearly  independent  functions 
7t', ,  .  .  .  ,  zv^>'  which  satisfy  the  relation  (74),  and  every  other  function 
satisfying  this  relation  is  a  linear  function  of  {Uf  Vf  w).  For  these 
functions  zc  we  have 


(75) 


Sw^  =  s^zi\  +  r, ,  .  .  .  ,  5mv"  =  s^w^o  4-  Ty>, . 


There  can  exist  no  linear  relations  between  the  functions  T,  of  the 
form 


{76) 


w  + 


+  C."n.,  =  U; 


where  C,  ,  •  •  •  ,  C"  are  constants,  and  f/  is  a  linear  function  of 
?/, ,  .  .  .  ,  ?(^;  for  if  there  could  be  such  a  relation,  it  would  also 
be  possible  to  find  a  linear  relation  connecting  the  functions 
w,  ,  .  .  .  ,  Wyi',  say  IV,  alone  and  satisfying 


(77) 


SW=s,W^U' ; 


where  C/' is  a  linear  function  oi  7i^,  .  .  .  ,  ?/^.  But  by  hypothesis 
the  linearly  independent  functions  v^,  .  .  .  ,  v^'  are  the  only  ones 
satisfying  such  a  relation,  and  consequently  we  can  have  no  such 
relation  as  (76).  It  follows  also  that  v"  cannot  be  greater  than  v' , 
and  consequently  cannot  be  greater  than  v. 

Suppose  now  that  v -\- v'  -\-  v"  =  \\  we  can  obviously  without 
loss  of  generality  replace  T^,  .  .  .  ,  F^"  hy  i\  ,  .  .  .  ,  tv,  etc.  We 
find  then  that  the  integrals  corresponding  to  the  root  s^  of  the  char- 
acteristic equation  divide  into  v"  sub-groups  of  three  elements  each, 
v'  —  v"  sub-groups  of  two  elements  each,  and  v  —  r'  sub-groups  of 
one  element  each.     Denote  these  by  III,  II,  I,  respectively  ;  then 


(78) 


'III    =^    (7^,,    Z\,    W^  ...   (?V,    "Vv",    "^v"), 

I  =  (?V  +  i)  •   .    .  {^(^), 


SUB-GROUPS  OF  INTEGRALS.  83 

giving  the  relations 

'  from  III,  Sii^  =  s^u^ ,  Sv^  =  s,i\  +  ?/, ,  Szu^  =  s^za^  -f-  v^ , 

Stiv"  =  s^Uv'> ,  Sv^,"  ^s^Vy"  -\-  iiv" ,  Swy'i  =  s^w^ii  -j-  v^'i ; 
.(79)       \  from  II,    Su^»  + 1  =  J,?/,//  +  , ,  5:V"  + ,  =  ^I'ZV  +  z^^" , 

Suyi  =  SjUy' ,    Svy'  =  SjVy'  -f-  i(y> ; 
from  I,      Sh^'j^i  =  ^;?/^'_l_i  ,   .  .  .  ,    Sn^  =  sji^. 

By  a  continuation  of  this  process  we  find  a  series  of  numbers,  v,  v' , 
v",  .  .  .  ,  v*^',  such  that 

V  +   r'  +  .     .    .   +   r'*'    :=   A, 

and  where  no  r  is  greater  than  the  preceding  one.     For  y^^)  we  shall 
liave 

1^'*'  sub-groups  of  /^  -|-  I  elements  each, 

v<*  -  ''  —  !/<*'  "  "  /^  elements  each. 


V    —   V 

y    —  v' 


I  element  each. 


The  elements  of  a  group   containing  m  elements  are,  say,  7, ,  y^ , 
.  .   .  ,  y„i ;  then 

Sy^  =  -^1  Ji »     Sy^  =  s,y,  -{-y,,     .  .  .  ,     Sy„,  =  s,y„,  +  y,„  _  ^ . 

The  substitution  5  corresponding  to  the  point  x  =  o  now  takes  the 
form 

^1,12,       .    .    .    ,    Ja  ;      S,y^    .     -y,  J.    +  J,   ,  •    .    .    ,     S,  J/„  -f  ;^a  -  I 


(80)  5  = 


^1    >      ^2    J 


|;(fl   .       c     1/   ('■)         C      r    ('■)    -I-     1/   (fl  0      l/Cl   _|_    1/('') 


84 


LINEAR  DIFFERENTIAL   EQUATIONS. 


Jordan  in  his  Cours  d' Analyse  (vol.  iii.  p.  175)  gives  S  a  slightly 
different  form  and  one  which  is  a  trifle  more  convenient.  Jordan's 
form  is  derived  simply  from  (79),  viz.  :   Write 

y^    =  y.^    K    =  s.y. »    y.   =  -^.^s ,  .  .  .  ,    F„   =  .y,"  -  % , 
y  —  V  '     Y  '  —  s  V  '     V  =  s  V  '  Y ,  =  s'^'  -  'V  ' 


7      —   "        7 


c  '■-  7 

3     —  •'2  -3  >    •    •    •    >   -^/s 


'H 


We  have  now 


(80')     5  = 


Y      Y 

Y'    Y'  . 


;  s,Y,,  ..,(n+  F.),  ...,  5,(K+  F„_,j 

;-y.F/,  ..,(f;+f/),...,  .,(f„.+  f.,_o 


Z,  ,  Z,  ,  .  .  .  ;  5,Z,  ,  .y,(Z,   +  Z,),  .  .  .  ,  ^,(Z^  +  Z^  _  ,) 


Before  going  on  with  this  subject  it  will  be  convenient  to  take 
up  the  case  of  differential  equations  with  uniform  doubly  periodic 
coefficients,  and  give  Jordan's  method  for  reducing  the  substitutions 
5  and  S'  above  referred  to  to  their  respective  canonicaX_forms. 
Assuming  /,U'),  •  •  •  ,  /«(-^')  as  a  system  of  fundamental  integrals 
and  letting  5  and  S'  denote  the  substitutions  arising  from  the 
change  of  x  into  x  -j-  go  and  x  -{-  00'  respectively,  we  have  seen  that 


{a) 
and 


S  = 


S'= 


/i ;  «'ii  /i  +  •  •  •  +  «'.« /« 


/, ;  «'«./  +  •  •  •  +  «'«»/« 


/>;      «''n/i    +  •  •  •  +  «''.»/» 


/n ;  «''«i/i  +  •  •  •  4- «'»»/, 


DOUBLY  PERIODIC  COEFFICIENTS.  85 

and  also 

{c)  SS'  =  S'S. 

We  can  by  a  method  entirely  similar  to  the  preceding  one  reduce 
the  substitutions  5  and  S'  to  their  canonical  forms.  Let  s  denote 
a  root  of  the  characteristic  equation  corresponding  to  S,  and  let 
y, ,  ^2,  •  •  •  denote  those  independent  integrals  of  the  differential 
equation  which  are  multiplied  by  s  when  the  substitution  5"  is  made, 
z.t\  when  x  is  changed  into  x  -\-  00.  The  general  form  of  the  inte- 
grals possessing  this  property  is  then 

«ij^i  +  «^2j2  +  .  .  .  . 

Suppose  that  on  applying  the  substitution  S'to  y^we  change  7, 
into  Fj  ;  now  the  substitution  SS'  changes  y^  into  sV^,  and  S'S 
must  produce  the  same  result  ;  but  S'  changes  y,  into  F, ,  and  so  S 
should  change  V,  into  5  F,;  it  follows  then  that  F  must  have  the 
form 

«^i  ji  +  ^.y.  +  .  .  .  . 

The  substitution  S'  thus  replacing  each  of  the  integrals  jj ,  y^ ,  •  •  • 
by  linear  functions  of  these  same  integrals,  there  must  exist  at  least 
one  linear  function,  say  71,  of  these  integrals  which  S'  changes  into 
j'iL  We  have  thus  shown  that  there  exists  at  least  one  integral,  u, 
which  the  substitutions  5  and  S'  change  into  su  and  s'ti  respec- 
tively. 

We  proceed  now  to  show  that  it  is  always  possible  to  find  a  set 
of  fundamental  integrals,  say 

Jii  »   •  •  •  ,   III,  ,     J21  ,    •  •   •  ,  ^2/2  ,      .   .  .  ,  7ai  ,    •   .  •  ,  Ja/^, 

C*  r»  «*  r*  c  r*     J 

<i>II     ,     .      .     .     ,      '-'iw;,    »      '^21     >       •      •     •     )       '^2H;j    >       •      •     •     >       —  /HI    )     •     •     •     >  ^'jii    ' 


such  that  the  substitutions  5  and  S'  shall  take  the  forms 

yik,  . . .  ,  y;k,  .  •  • ;   s.y,/, ,  . . . ,  .jXj/a  +  ^ik),  • 

(d)  S  =      s,k  ,  .  •  .  ,  2i/,,  .  .  .  ;    s,s,!,  ,  .  .  .  ,  slsik  +  Zik),  . 


86  LINEAR  DIFFERENTIAL  EQUATIONS. 

jiA ,  . . . ,  J.A ,  •  •  •  ;  s^y^k ,  •  ■ ' ,  s/{j^ik  +  Y'ik), . . . 
(r)  S'=     2,kj  .  • ' ,  Zik,' ' ' ;  sjz.k ,  ' '  ■ ,  s^{2ik  +  z'ik), . . . 


where  {s, ,  s/),  {s^,,  sj),  ...  are  pairs  of  different  constants,  i.e.,  s,  is 
never  equal  to  s.,,  s/  is  never  equal  to  sj,  etc.;  and  where  F,*,  F.^ 
are  linear  functions  of  the  integrals  y  whose  first  sufifix  is  less  than 
t ;  Zjk ,  Z' ik  are  linear  functions  of  the  integrals  z,  whose  first  sufifix  is 
less  than  i\  etc.  Assuming  this  proposition  true  for  substitutions 
containing  less  than  ii  variables,  we  will  prove  it  to  be  true  of  the 
substitutions  5  and  S'  containing  n  variables.  We  have  seen  that 
there  always  exists  an  integral  ?/  which  is  changed  into  su  and  s'u 
by  the  substitutions  5  and  S'  respectively.  In  the  system  of  funda- 
mental integrals  /,  Z^,  .-•,/»  let  us  replace  any  one,  say/„ ,  by 
the  integral  u.     The  substitutions  5  and  S'  then  take  the  forms 

(/)  5  =  1  /,,...,/«-■,  ?^ ;  ^.  +«i  u,  .  .  .  ,  F„_,-\-  a„  _  ,u,  su  I  , 
(g)  5'  =  1  /,...,/«-  X ,  ?^ ;  ^'.+«/«,  .  .  .  ,  F'„_  ,+«'„  _ ,?/,  s'ti  I  , 
the  functions  Fi  and  F-  denoting  linear  functions  of /,...,/„_  j . 

Consider  now  the  substitutions 
(/f)  ^  =  I  /i  ,    .••>/«-!>     F^  ,    .  .  .  ,    !"„  -  I  \  , 

(z)  -S"  =  1  y"i ,   .  .  .  ,  y^  -  I  ;     F^  ,   .  .  .  ,  r  n-  1  \  f 

q{  n  —  I  variables.     From  the  relation 

SS'  =  S'S 

follows  at  once 

(y)  22'  =  2' 2. 

Applying  now  the  above-stated  theorem  to  these  substitutions,  they 
may  be  written  in  the  forms  {d )  and  (r),  fi  —  i  variables  appearing 


DOUBLY  PERIODIC  COEFFICIENTS. 


87 


instead  of  n.     This  same  change  of  independent  integrals  will  put 
5  and  S'  in  the  forms 


{k)    S 


(/)    S'  = 


y.k,  . 

■  ■,yik,  . 

■  • ;  sj,k+c,tu,  . 

.  ,   •«■.(  J:A-+  yik)+Cik7t,     .   . 

^ik  >  • 

•  •■>   ^ik  J    • 

.  . ;  s^,k-^d,i:U,  . 

..siZik^Zi^^dikU,      . 

u 

;  su 

y^k,  • 

..,/,•*,     . 

•  • ;  -Ji jiA+^i*?^  . 

•,-^:u-.+  n)+^>, .. 

^ik,  • 

•  ■•  J  ^ik  •     • 

•!    •^2'3'iA~T""]/t^^)    • 

.  -,  s',{3i^-\-Z-k)-{-d;,n,  .. 

S  11 


Suppose  now  we  replace  the  independent  integrals  yn,  by  the  follow- 
ing: 


{m) 


y'ik  =  yik  +  ocikti ; 


the  substitutions  6"  and  S'  will  retain  their  forms;  but  the  constants 
Cik  and  c'ik  will  be  changed  into  [r,^]  and  [c'^^,  where 


{0) 


i/,-4  and  i/i^  denoting  what  F^-^  and  F/^  respectively  become  when  in 
them  we  replace  the  functions  j  by  the  corresponding  constants  a. 

We  will  now  first  assume  that  s  is  not  equal  to  s^  ;  in  this  case 
we  can  clearly  assign  such  values  to  the  constants  a  that  all  the  new 
coefficients  [r^^]  shall  vanish.  It  is  easy  to  see,  by  aid  of  the  relation 
SS'  =  S'S,  that  the  vanishing  of  these  constants  will  involve  the 
vanishing  of  the  constants  [t',-*].  Equating  the  coefificients  of  2(  in 
the  expressions 

SS'yik         and         S'Sjfij^, 

and  denoting  by  7"^  andF/^  what  F,-4  and  F,4  become  when  the  inte- 
grals y  are  replaced  by  the  corresponding  constants  c  and  c',  we 
have 


(/) 


s,{c',,  +  r,,)  +  /^,,  =  s/{c,,  +  r,,)  +  sc'a- 


\ 


88  LINEAR  DIFFERENTIAL   EQUATIONS. 

If  now  the  constants  Cik  are  all  zero  these  relations  (/)  reduce  to  the 
form 

{q)  {s,  —  s)c\k  +  s.r^k  =  o. 

These  equations  {g)  are  linear  and  homogeneous  in  the  quantities 
c'i,,,  and  their  determinant  is  a  power  of  s^  —  s  ;  but  since  by  hypoth- 
esis s,  is  not  equal  to  s,  this  determinant  cannot  vanish,  and  there- 
fore, in  order  that  equations  {q)  may  be  satisfied,  we  must  have  all 
of  the  quantities  c'^^  equal  zero.  In  the  same  way,  if  s'  is  not  equal 
to  s^'  we  can  make  all  the  constants  c' ^^  vanish,  and  their  vanishing 
will  also  involve  the  vanishing  of  all  the  constants  c,i. .  Continuing 
this  process,  suppose  that  none  of  the  relations 

are  satisfied ;  then  we  can  make  all  of  the  constants  Ci^ ,  c'ik ;  di^ ,  d'n, ; 
.  .  .  ,  disappear,  and  so  the  substitutions  5  and  S'  will  be  in  the 
canonical  form,  and  to  the  different  classes  of  integrals  j',  z,  .  .  .  we 
have  added  the  class  composed  of  one  integral  only,  viz.,  u. 

Suppose  now  that  s  ^  s^ ,  s'  =:  s/;  as  before  we  can  make  all  the 
coefificients  ^/^ ,  d'^/, ;  .  .  .   ,  vanish.     If  now  we  write 

<^,k  =  s,  Yik ,  c'ik  =  s.'y'ik , 

we  will  again  have  the  substitutions  5  and  S'  in  the  normal  form, 
the  new  integral  7i  entering  now  into  the  category  of  integrals  ji^ 
belonging  to  the  class  of  integrals  y,  which  have  unity  for  their  first 
sufifix,  i.e.,  the  class  which  the  substitution  S  multiplies  by  s,  and 
which  the  substitution  S'  multiplies  by  s'. 

We  resume  now  our  original  problem  of  determining  the  forms 
of  the  integrals  of  the  linear  dilTerential  equation  with  uniform 
coefficients.  Starting  from  equation  (8o'),  Hamburger  proceeds  to 
determine  the  forms  of  the  integrals ;  but  Jordan  gives  a  briefer  and 
rather  more  elegant  shape  to  Hamburger's  method,  so  we  shall  em- 
ploy it.     Write  as  before 

!^f  =  r 
2m 


FORMS  OF   THE  INTEGRALS.  89 

{dropping  the  subscripts  for  convenience).  The  substitution  5  being 
in  the  canonical  form  (80'),  let  y^,  y^  ,  -  •  •  ,  yk  denote  a  group 
of  integrals  to  which  S  applies,  and  let  s  denote  the  corresponding 
root  of  ^  =  o.     Write 

y,  =  x*-s, ,    y,  =  x^s, ,  .  .  .  ,  y^=  x^Zk. 

Since  after  once  turning  round  ;r  =  o,  x""  reproduces  itself  multi- 
plied by  e'^'"'*' ,  =  JT,  it  follows  that  the  functions  z  submit  to  the  sub- 
stitution 

(81)  '2=\z,,  .  .  .  ,Zk\     z,,  .  .  ,  ,  Zi-\-Zi_,\  ; 

since 

Sy,  =  sy, ,     Sy,  =  s  {y,  +  j„),     .  .  .  ,     Sy^  =  s  {y^  +  j'^  _  ,)• 

It  is  only  necessary  now  to  find  the  forms  of  ^„ ,  ;;j ,  .  .  .  ,  Z/,. 
The  first, -jj ,  is  obviously  a  uniform  function,  since  Sz^^  z^.  To 
get  the  forms  of  -sr^ ,  .  .  .  ,  ^•^  we  introduce  a  new  function,  6^ ,  de- 
fined by 

/o  N  -         log  X 

(82)  e  =-^-^. 

'  2711 

The  effect  of  turning  once  round  x  =  o,  that  is,  of  applying  the 
substitution  S,  is 

(83)  Sd,=  d,Jr^- 

Introduce  now  the  series  of  functions  6^1,  6^,  .  .  .  ,6^4,  defined  by 
the  equations 

(84)     e='^ g,(.. -.)...(.-.+  .) 

^  ^^         •        2m    '  '  \  .2  .  .  .  k 

for  these  we  have,  as  the  result  of  turning  once  round  x  ^=0, 

(85)     5«.=  ..  +  . ^,,^ffl+.)^,W-.)...j^,-^-+^). 


90  LINEAR  DIFFERENTIAL  EQUATIONS. 

Add  and  subtract  6k  in  Sd^ ;  then  for  the  right-hand  member  we  have 


(86)  e, 


\  .2  .  .  .  k 


\  .2  .  .  .  k 

e^{8,-i)..,{e,-k^2) 

\  .2  .  .  .  k 


=  ^*+ 1.2.../^-! -^*+^*-:- 

We  have  then  finally  for  the  functions  Q, 

(87)      Sd,  =  ^.  +  I,    se,  =  e,-\-e,,    .  .  .  ,    se,  =  e,  +  d,_. 

The  result  of  A  turns  round  -r  =  o  gives 


(88) 


S'6^=d,  +  X,     S'H,=  d,  +  Xd,+ 


A  (A -I) 


I  .  2 


5^^^=^^+A^^_,  +  ^^^^-^-^'^^_,+  .  .  .  + 


If  A  =  /(',  this  last  is 

(89)      s'^-e,  =  6,  +  kdk-.  +  ^^\ ~  '^ ^^_,  +  .  .  .  +  I. 

If  A  = -^ -}- /,  the  coefficients  in  this  equation  change  into  the 
binomial  coefficients  corresponding  to  the  exponent  {k  +  /)  ;  as 
there  are  no  functions  such  as  B_i{i  a  positive  integer),  the  last 
term  is  simply  the  binomial  coefficient 

{l^k){l^k-X)    .    .    .    (/+!) 

If  now  we  choose  a  system  of  uniform  functions  u^,  ti^,  .  .  .  ,  iik  , 
we  can  write 


(90) 


And  these  values  obviously  satisfy  the  condition  (81). 


FORMS  OF   THE  INTEGRALS.  QE 

We  have  now  for  the  integrals  jj/j, ,  .  .  .  ,  yk  the  same  forms  as. 
those  in  equations  (49) ;  retaining,  however,  Jordan's  notation,  we 
have 


(91) 


f  J„    =    XrM,  , 


The  uniform  functions  yI/„ ,  .  .  .  ,  M^,  N^ ,  .  .  .  ,  N^  .  .  .  are 
obviously  linear  functions  of  the  A  -{-  i  independent  functions 
;/(,,?/;,...,  7(i  of  equations  (90),  and  in  particular  M^ ,  Jl/^ ,  .  .  .  ,  M^ 
differ  only  by  constant  factors  from  each  other  and  from  //„ .  It 
follows  from  this  that  the  functions 

x^M,,     x^M,,     .  .  .  ,     x-3fi, 

which  differ  only  from  x''M„  by  constant  factors,  are  integrals  of 
the  differential  equation.  These  functions  3f  and  N  are  so  far 
perfectly  arbitrary  uniform  functions ;  they  will,  however,  in  par- 
ticular cases  be  seen  to  divide  themselves  into  two  classes — one 
containing  only  a  finite  number  of  negative  powers  of  x  (or  of  ;ir  —  a, 
if  a  be  the  critical  point  considered)  and  one  containing  an  infinite 
number  of  such  powers. 

When  a//  of  the  functions  M,  N  entering  into  any  one  of  the 
integrals  of  the  equation  contain  only  a  finite  number  of  negative 
powers  of  the  variable  x,  the  integral  is  said  to  be  regular  in  the 
region  of  the  point  x  =  o.  A  very  important  class  of  equations,  first 
investigated  by  Fuchs,  is  that  class  in  which  all  of  the  integrals  in  the 
region  of  a  critical  point  are  regular.  The  investigation  of  this  class 
will  be  given  in  the  following  chapters. 

The  substitution  5  which  we  have  been  considering  is  of  course 
only  one  of  a  number,  finite  or  infinite,  which  belongs  to  the  linear 
differential  equation  with  uniform  coefficients.  If  the  variable  be 
made  to  describe  all  possible  paths  enclosing  one  or  more  of  the 
critical  points  a,  b,  c  .  .  .  oi  the  equation,  we  shall  have  a  certain 
substitution  corresponding  to  each  of  the  paths  ;  the  aggregate  of 
all  these  substitutions  is  called  Xhe:  group  of  the  equation.  We  will 
denote  this  group  by  the  letter  G.     It  is'clear  that  G  will  assume 


'92  LINEAR  DIFFERENTIAL   EQUATIONS. 

-different  forms  according  to  the  choice  of  the  system  of  fundamental, 
i.e.,  hnearly  independent,  integrals.  This  notion  of  the  group  of  a 
linear  differential  equation  is  of  the  highest  importance  in  the  theory, 
and  will  be  treated  of  more  fully  in  another  chapter  of  this  volume, 
and  still  more  fully  in  Volume  II.  From  a  knowledge  of  the  group 
of  an  equation  we  can  derive  all  the  essential  properties  of  the  equa- 
tion. Suppose,  for  example,  /*=  o  is  an  equation  having  all  of  its 
integrals  regular:  it  is  obvious  that  among  equations  of  this  type 
are  included  all  equations  having  only  algebraic  integrals.  What, 
then,  are  the  conditions  which  P=  o  must  satisfy  in  order  that  all 
of  its  integrals  may  be  algebraic  ?  It  is  obviously  necessary  that  the 
different  functions  into  which  the  substitutions  of  the  group  G 
change  the  chosen  system,  say  jj,  j^*  •  •  •  ,  J« »  of  fundamental 
integrals  must  be  limited  in  number,  and  consequently  it  is  sufficient 
that  the  group  G  contain  only  a  finite  number  of  substitutions. 
The  case  of  these  equations  will  also  be  returned  to  later  on. 
-Among  the  substitutions  which  enter  into  a  group  there  are,  since 
we  consider  only  the  case  of  a  finite  number  of  critical  points,  only 
a  finite  number  of  independent  ones.  Suppose,  for  example,  that  5 
is  a  substitution  corresponding  to  a  given  closed  contour  ii',  and  that 
vS",  is  another  belonging  to  a  given  closed  contour  K^ ;  then,  if  we 
describe  successively  the  two  contours  K  and  K^ ,  we  shall  arrive  at 
a  substitution  55,  which  is  the  resultant  of  the  two  substitutions  5 
and  5, .  All  of  the  substitutions  in  a  group  therefore  result  from  the' 
combinations  which  can  be  made  among  the  substitutions  belonging 
to  each  of  the  critical  points  taken  separately. 


CHAPTER    IV. 

FROBENIUS'S   METHOD. 

We  will  consider,  as  before,  the  region  of  the  critical  point  ;t:  =  o 
and  let  PJ,x),  PJyx),  .  .  .  ,  P„{-x:)  denote  convergent  series  proceeding 
according  to  positive  integral  powers  of  x,  and  further  assume  that 
Pa{x)  does  not  vanish  for  ^  =  o.  If  all  the  integrals  of  the  given 
differential  equation,  P{y)  =  O,  are  regular  in  the  region  of  x  =  o, 
the  equation,  as  will  subsequently  be  seen,  can  be  put  in  the  form 

(I)        Plx)x-  ^  +  Pix)x'^  -  >^;r^  +  .  .  .  +  P.{x)y  =  o.       '' 

We  will  now  seek  to  determine  the  form  of  the  integrals  of  this 
equation  by  an  extremely  elegant  and  ingenious  method  due  to 
Frobenius.*  For  simplicity  we  will  assume  PJ^x)  =  i.  In  equation 
(i)  make  the  substitution 


(2)  y=g{x,r)  =  2g,x-^\ 

1/  =  o 

The  limits  r  =  o  and  v  ^  co  will  hereafter  be  omitted  from  the 
summation  sign,  but  will  be  always  understood.  We  see  at  once 
that  this  substitution  gives  rise  to  the  equation 

(3)  Pi^g.x'-  + ")  =  2o^P{x^  + "). 


*  Ueber  die  Integration   der  linearen    Differentialgleichungen   durch  Reihen.     (Von 
Herrn  G.  Frobenius.)     Crelle,  vol.  76. 

93 


•94  LINEAR  DIFFERENTIAL   EQUATIONS. 

If  now  we  write 

(4)         f{x,  r)  =  r(r  -  I)  .  .  .  (r  -  «  +  \)Plx) 

^r{r-x)...{r-n-^  2)Pix)  +  .  .  .  +  P„(^), 


P{x-)  =  x^f{x,  r\ 


we  have 

<5) 

and  consequently 

.(6)  P\_g{x,  r)\  =  ^g.f{x,  r  +  v)x^^\ 

Since  the  functions  Pipe)  are  developable  in  convergent  series  going 
according  to  positive  integral  powers  of  x^  it  follows  that  the  series 


<7) 


f{x,r)  =  2/.{r)x'' 


is  also  convergent,  and  that  the  coefificients  of  the  different  powers 
of  X  are  integral  functions  of  r  of  the  degree  n  at  most.  The  sub- 
stitution jj/  =^  jr{x,  r)  being  made  in  (i)  gives  us  now 

(8)     ^[g^f{r+  y)  +  g.  -  .fir+  V  -  i)+  .  .  .+.-./.-. (r+ I) 

In  order,  then,  that'j  =  g{x,  r)  shall  be  an  integral  of  (i),  we  must 
have 

g,/{r)  =  O, 

(9) 


g^Ar+y)+g.-rMr+y  -  i)+.  .  ■+.gj.-r{r+  i) 

+  goMr)  =  o. 

If  we  suppose  now  that  ^^-r^'is  the  first  term  in  the  series 

g{x,  r)  =  ^g^x^-^", 

then  g^    cannot    be    zero,  and  consequently  r  must    be    a  root  of 
the  equation  f{r)  =  o,  which  is  of  the  «^''  degree   in   r.     In  what 


FROBENIUS'S  METHOD.  95 

follows  we  will  consider  r  as  a  variable  parameter,  and  g^  (or  simply 
g  for  convenience)  as  an  arbitrary  function  of  r.  Neglecting  the  first 
of  equations  (9),  we  can  at  once  determine^,,  g^,  ...  3,5  functions 
of  r.     Write 


(10)    (-iWr)  = 


flr^y-i),flr+y-2),  . .  .,/._,(r+i),/,(r) 
fir4-y-i),flr-^ry-2), . . .,/..,(/-+ i),/..X^) 
o,  /(^+^-2),  ...,/. -3(^+1), /,.,(r) 


o,  O,  ...,/(r+i),      fir) 

now  from  (9)  and  (10)  we  have 

g{r)h,  {r) 


(11)  g^{r)  = 


f{r-\-i)f{r-\-2)...f{r+v) 


The  variable  parameter  r  will  be  so  restricted  that  all  of  its  values 
shall  be  found  in  the  regions  of  the  roots  of  the  equation /"(r)  ^  o. 
Since  the  roots  of  this  equation  have  their  moduli  all  less  than  a  cer- 
tain determinate  finite  quantity,  it  is  easy  to  see  that  these  regions 
can  be  chosen  so  small  that  the  denominator  of  the  rational  func- 
tion ^^(r)  shall  only  vanish  for  the  roots  of  the  equation  f{r)  =  o. 
This  vanishing  of  the  denominator  of  gv{r)  would,  however,  make 
^„(r)  infinite  in  general ;  this  difificulty  can  nevertheless  be  avoided 
by  a  proper  choice  of  the  arbitrary  function  g{r).  We  will  suppose 
that  the  roots  of  f{r)  =  O  are  arranged  in  groups  in  the  manner 
described  in  the  last  chapter,  and  suppose  further  that  e  (necessarily 
an  integer)  is  the  maximum  difference  of  two  roots  in  any  of  the 
groups.     Writing  now 

(12)  g{r)=f{r+  i)/(r+  2)  .  .  .  /(r  +  e)C{r), 

where  C  {r)  is  an  arbitrary  function  of  r,  it  follows  that  for  all  the 
values  of  r  under  consideration  the  functions  gv{r)  are  finite,  and 
consequently  that  if  the  series  y  =z  g{x,  r)  is  convergent,  then 
y  —  gi'^y  ^)  is  an  integral  of  the  differential  equation 

(13)  ^{y)  = /{r)g-{r)x': 


96  LINEAR  DIFFERENTIAL  EQUATIONS. 

We  have  now  first  to  investigate  the  conditions  for  convergence  of 
the  series 

g{x,r)  =  ^g.x-'r-. 

If  we  assume  r  >  e,  then,  recalhng  the  definition  of  e,  it  is  clear 
that/(r  -\-  V  -\-  i)  cannot  vanish  for  any  of  the  values  of  r  to  which 
we  are  restricted,  and  so,  from  equations  (9),  we  have 

(14)  ^.+,=  -  y(^  ^  ^  ^  AsJSr  -\-y)^-g^- 1/.(^+  ^'  -  I) 

+  •  •  •  +.-/.+ X^)]- 

Denoting  now  by  F^{r)  and  G^{f)  the  moduli  of  /„(r)  and^'-^(r),  we 
have  at  once  the  inequality 

(15)  G^.  +  :  ^  T{^^rJ^)\-^^^^^'-^  ^)+  G...Fir^y  -  i) 

+  .  .  .  +6^/^.+  ,(r)]. 

We  will  now  suppose  a  circle  of  radius  K  drawn  with  the  point 
jtr  =  O  as  centre,  and  where  A' is  as  little  less  as  we  please  than  the 
radius  of  a  circle  inside  of  which  the  functions  P^{x),  PJ^x),  .  .  .  , 
Pn{x)  are  all  convergent ;  also  let  f\x,  r)  denote  the  derivative  of 
fix^  r)  with  respect  to  x.     The  series 

f{x,  r),     =  ^f,{r)x\ 
and 

f\x,  r),     =  2{v  +  i)/^^,{r)x% 

are  both  convergent  so  long  as  the  inequality 

mod.  X  <  K 

is  satisfied ;  I.e.,  so  long  as  the  point  x  remains  inside  the  circle  of 
radius  K,  or,  we  may  say  for  brevity,  so  long  as  x  remains  inside  the 
circle  K.  Let  now  M{r)  denote  the  maximum  value  which  the 
modulus  oi  /'{x,  r)  takes  on  the  circumference  of  the  circle  K\  then 
by  a  well-known  theorem  we  have 

(16)  i^.  +  xW<  -^r^^M{r)K-^<M{r)K-% 


FROBENIUS'S  METHOD. 
and  consequently,  from  (15), 


97 


(17)       G^^^<y^_^j^^\_G.M{r^v)-^G._^M{r^v-i)K-^ 

+  .  .  .  -^GM{r)K-'\ 

Denoting  the   right-hand   member  of  this  inequahty  by  «„  +  i,  we 
have 


(18) 


a^j^i  — 


GJI{r  +  v)  a„F{r  +  v) 


or,  since  G"j,  <  «^ , 
(19)  «,  +  !  <a^ 


^(^+^+1)       A'F(r+y-fi)' 


~    M{r-^v)  Fir  -\-  v)      - 

_F(r+r+  i)  +  A^/^(r  +  1/  +  i)_ 


Still    assuming    y  >  e,  we  will  define  certain  quantities  b^  by  the 
formula 


(20) 


(^^  + 1  =  <5^ 


i^/(r+r)  F{r^v) 


L/^(;'+y+i)   '    A"/^(r  +  ?- +  i)_ 


also  assume   b^,  as  we  obviously  can,  so   that   we  shall  have  the 
inequalities 


(21) 


Gv  <  ttv  <b^. 


We  know  that  the  integral  function  f{r)  is  of  degree  n  in  r,  and 
therefore  when  v  increases  indefinitely  the  quotient 


and  of  course  its  modulus. 


f{r  +  v) 
F{r  +  v) 


tends  to  the  limit  unity.  Further:  we  have  assumed  PJyX)  =  i,  and 
consequently /'(;ir,  r)  is  an  integral  function  of  r  of  degree  at  most 
equal  to  n  —  i  ;  also,  M{r)  denotes  the  maximum  value  which  this 
function  can   have   when  mod.  x  =  K.      It  is  now  easy  to  see  (the 


98  LINEAR  DIFFERENTIAL  EQUATIONS. 

rigorous  proof  will  be  given  immediately)  that  if  v  increases  indefi- 
nitely we  have 

(22)  hm.  C-,     I —    IN  =  o. 
^     '  F[r-\-  v-\-  \) 

It  follows  now  from  (20)  that 

(23)  l.m.  -^-  =  ;^, 

and  therefore  that  the  series  ^b^.x",  and  therefore  also  the  series 

<24)  g{x,  r),     =  :^g,{f)x-+\ 

is  convergent  inside  the  circle  K. 

It  is  necessary  now  to  show,  by  aid  of  the  results  already  ob- 
tained, that  the  series  (24)  is  uniformly  convergent  for  each  of  the 
values  of  r  under  consideration.  If  we  denote  by  d  a  given  arbi- 
trary small  quantity,  we  must  show  that  for  all  of  the  considered 
values  of  r  it  is  possible  to  find  a  finite  number,  say/,  such  that  the 
modulus  of  the  sum 

1/  =  00 

V    =J 

shall  be  less  than  6.  In  establishing  this  we  will  first  prove  the 
truth  of  equation  (22).     Let  s  denote  the  modulus  of  r;  i.e., 

s  =  mod.  r  =:  \  r  \  . 

(The  symbol  |  A''  |  ,  due  to  Weierstrass,  will  be  used  when  con- 
venient to  denote  the  modulus  of  the  quantity  X,  whatever  X  may 
stand  for.)  Also,  let  M^ ,  M^ ,  .  .  .  ,  M„  denote  the  maximum 
values  of  the  moduli  of  P/ix),  P^'(x),  .  .  .  ,  P,/{x)  on  the  circum- 
ference of  the  circle  K;  if  then  we  write* 

(25)         iis)  =  ^(^  +   I)  .  .  .  (^  +  ;/  -  2)Af, 


*  For  the  moment  we  will  allow  r  to  vary  indefinitely  until  it  is  necessary  to  re-intro- 
,duce  ;^  +  r  as  the  indefinitely-increasing  argument. 


FROBENIUS'S  METHOD.  99 

we  have 

(26)  M{r)  <  tp{s). 

The  function  y(r)  is  of  degree  n  in  r,  and  we  may  thus  write 

/{r)  =  r"-\-  [/(r)  -  r«], 
and  so  obtain  the  inequaHty 

^{r)  ^  ^"  -  I  /{r)  -  r"\; 

•again, 

/(r)  =r{r  —  i)  .  .  .  {r  —  n  ^  i) 

+  PXo)r{r  -  I)  .  .  .  (r  -  ;^  +  2)  +  .  .  .  +  P„{o). 

Let  n,,  .  .  .  ,  n„  denote  the  moduh  of  P,{o),  •  •  ■  ,  Pn{o) ;  then, 
if  we  write 

cpis)  =  5(5  +  I)  .  .  .  (^  +  ;/  -  I)  +  n,s{s  +  I)  .  .  .(^  +  ;,  -  2) 

we  have 

{27)  I  f{r)  -r"\  <  cp{s), 

and  consequently 

(28)  F{r)  >  s"  -  (P{s), 

provided  only  we  choose  j-  so  large  that  the  right-hand  side  of  this 
inequality  shall  be  positive ;  this  can,  of  course,  always  be  done, 
since  (p{s)  is  only  of  degree  n  —  1.  From  the  inequalities  obtained 
above  we  derive  at  once 

_  M{r  +  v)  V;  \  r  -}-  r  \     

^"^^  F{r  -{-  V+  1)  ^  \  r-^y+  i~f «~-  0  |  r  +  k  +  i  |  ' 
but 

I  r  -f-  ^  I  <  ^  -f-  "^  and           \r-\-r-\-i\'>v  —  s; 


lOO  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  since  the  positive  functions  rpi^s)  and  s"  —  (f){s)  always  increase- 
after  a  certain  value  of  the  argument  s,  we  have,  by  giving  v  a 
sufficiently  large  value, 

M{r^v)  iiv  +  s) 


F{r  +  ^  +  i)        (^  —  -$■)"  —  0(''  —  s)' 

The  numerator  of  the  right-hand  member  of  this  inequality  is  of 
lower  degree  than  the  denominator,  and  so  this  member  tends  to 
zero  as  v  increases  indefinitely  ;  that  is,  we  have,  as  above  stated. 

For  the  completion  of  the  proof  of  the  convergence  of  the  series 
g{x,  r),  we  notice  that  since  all  of  the  roots  oi  f{r)  =  o  lie  in  a  finite 
region,  and  since  the  parameter  r  can  only  vary  in  the  regions  of 
these  roots,  then  s  must  be  always  Jess  than  a  certain  determinate 
quantity,  say  /.  If  now  we  take  v  sufficiently  large,  we  have  ob- 
viously 

V3-J  C-(^  _i_   „  _L   i\    "^ 


F{r  -{-  y  -\-  I)        {v  —  t)"  —  (p{y  —  t) 
and 

F{r+v)  (1^  +  /)" +  0(^  +  0 

^^^>  Fir  +  r  +  i)  ^  (k  -  ty  -  (P{v  -  t) ' 

Suppose  y  =  /<  is  the  value  of  v  for  which  these  inequalities  first 
hold;  they  will,  rt/c^r//!?/-/,  continue  to  hold  for  all  values  of  v  greater 
than  1,1.  Suppose  now  quantities  c^  to  be  defined,  for  all  values 
of  V  equal  to  and  greater  than  /<,  by  the  formula 


)34)       ^v-\-i  —  <^v 


_{y  —  ty—  (p{v  —  t)  ^  K  {y  —  ty  —  (p{v  —  /)J  ' 


then  if  we  choose  c^y  b^,  as  we  obviously  may,  we  have  generally 
0>  bv     Now  if  k  be  chosen  as  little  smaller  as  we  please  than  Ky 

then,  since   lim.  — — -  ^~^ ,  the  series  '2e^,k''  is  convergent.     Begin- 

ning  with  the  first  term  of  this  series,  and  counting  forward,  we  can 


FROBENIUS'S  METHOD.  \0\ 

cut  off  a  finite  number  of  terms  such  that  the  sum  of  the  remaining 
ones,  say  2  c^  k^ ,  shall  be  less  than  an  arbitrarily  chosen  quantity 

V-J 

6k  ~  ^  Now,  since  we  can  of  course  choosey'  greater  than  >u,  it  follows 
that  we  have 

\''ig,{r)x^'rv  I  <  s 

for  all  values  of  r  inside  the  regions  of  the  roots  oi  f(r)  =  o,  and  for 
all  values  of  x  inside  the  circle  of  radius  k  and  having  the  point 
-r  =  o  as  its  centre.     The  series 

g{x,r)  =  :Sg,x^+' 

is  therefore  uniformly  convergent  and  can  be  differentiated  with 
respect  to  r,  and  its  differential  coefficient  so  formed  will  be  equal  to 
the  sum  of  the  differential  coefficients  of  its  successive  terms. 

Having  established  now  the  convergence  of  the  series  g{x,r), 
we  proceed  to  investigate  the  forms  of  the  integrals  of  the  equa- 
tion P  (j)  =  o. 

We  will  consider  the  group  of  yw  -f-  I  roots  r^ ,  r^,  .  .  .  ,  r^  oi  the 
equation  f{^r)  =  o.  These  roots  are  so  arranged  (as  already  de- 
scribed) that  for  a  <.  §  the  difference  r^.  —  r^\s  a  positive  integer. 
As  certain  of  the  roots  of  this  group  may  be  equal,  we  will  further 
.assume  that  r„ ,  r,,,  r^,  Ty,  .  .  .  are  the  distinct  ones;   then 

^0    =  r,         =...=:  r^  _  X  will  be  an  o'-tuple  root  oi  f{r)   =  o; 
fa=   r<i  +  I  =  .  .  .  =  r^  _  X  will  be  a  (y^  —  a')-tuple  root  oi  f{r)  =  O  ; 
rp  =   rj3_^  I  =  .  .  .  =  /-y  _  ,  will  be  a  (;/  —  yS)-tuple  root  oi  f{r)  —  o  ; 

etc.  etc. 

We  have  assumed  for^(r)  the  form 

g{r)  =  f{r  +  i)/(r  +  2)  .  .   .  /(r  +  e)  C {r)  ; 

where  C{r)  is  an  arbitrary  function  of  r.    Recalling  now  the  definition 

of  e,  we  have  the  inequality  e  >  r^  —  r^,  and  therefore  g{r)  cannot 

vanish 

for  r  =  r„  =  r,         =  .  .  .   —   r^  _^\  but 

for  r  =  T-a  =  r^  +  1  =  .  .  .   =  r^  _  , ,  g{r)  is  zero  of  the  order  a, 

for  r  =  r^  =  r^  _[_  X  =  .  .  .   =  r.^  _  ^ ,  g{r)  is  zero  of  the  order  y5,  etc. ; 


I02  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  generally  for  r  =  rk,  g{r)  is  zero  at  most  of  the  order  ^.  Again  i 
the  expression  f{r)g  {r)x''  is  zero 

of  the  order  a  for  r  =■  r^  ^^  r^        =  .  .  ,  =  r^  _  , , 

of  the  order  /?  for  r  =  r„  =  ra  + 1  =  .  .  .  =  r^  _  i ,  etc.; 

and  generally  f{r)  g{r)x''  is  zero  of  at  least  the  order  k  -\-  \  for 
r  —Vk,  and  consequently  its  /^"'  derivative  with  respect  to  r  must 
vanish  for  r  —  r^.     Write  now  equation  (13)  in  the  identical  form 

and  differentiate  this  k  times  with  respect  to  r,  and  in  the  result 
write  Tk  for  r ;  we  have  as  the  result  of  this  operation 

(35)  P[^M-^.  ^^)]  =  0; 

d''g{x ,  r) 
where  g'^ix,  r)  =  — ^^^ . 

It  follows  at  once  that 

(36)  y=^''{^,^i) 

is  an  integral  of  the  equation  P{y)  =  o.     Since  the  series 

g{x,  r),    =  x'-:^g,{r)x'', 
is  a  uniformly  convergent  series,  the  same  is  true  of  the  series 

(37)  ^'{^,  r%    =  ^'''  ^  I  ^'Xrk )  +  kg' '  \rk )  log  x 

which  is  an  integral  of  P{y)  =  o.  Since  g{r)  vanishes  for  r  =  r^  at 
most  of  the  order  /',  it  follows  that 

g{fk),    g'{n),     •   .   .  ,    g^r^) 

cannot  all  be  zero,  and  so,  by  Fuchs's  definition,  the  integral  (37) 
belongs  to  the  exponent  r^ .  It  is  obvious  that  (37)  is  of  the  same 
form  as 

(38)  x'"'^\4y,  +  0,  log^  +  .  .  .  +  0^log*^|; 


FROBENIUS'S  METHOD.  IO3 

where  the  0's  are  uniform  and  continuous  functions  of  x,  and  are 
not  all  zero  for  x  ■=.  o.  Suppose  k  <  a\  then  from  (37)  we  find,  as 
the  coefftcient  of  \o^^x,  the  series 

(39)  x'k^glr^x, . 

Since  /^  <  a-,  we  have^(r^)  =^(r„),  and  so  this  series  cannot  vanish 
identically,  and  as  a  consequence  we  have  that  k  is  the  exponent 
of  the  highest  power  of  log  x  which  can  appear  in  the  integral 
g^ix^  Tk).  It  is  easy  to  extend  this  result,  and  so  observe  that  in 
general  the  integrals  which  belong  to  equal  roots  of/(r)  =  ohave 
different  exponents  for  the  highest  powers  of  log;ir  that  enter,  and 
consequently  that  these  integrals  are  linearly  independent.  In  par- 
ticular, if  r  is  an  «-tuple  root  oi  f{r)  =  o,  then  the  n  corresponding 
integrals  of  P{y)  =  O,  viz.,  g\x,  r^),  (/^  =  i,  2  ,  .  .  .  ,  n),  are  linearly 
independent.  From  what  has  been  said  it.  is  clear  that  the  form  of 
the  integral  ^'^(-r,  r^)  is  unaltered  if  we  add  to  it  a  linear  function 
(with  arbitrary  constant  coefficients)  of^'^~'(;ir,  r^_i),  .  .  .  ,g{^,fo)\ 
we  derive  from  this  fact  the  conclusion  that  the  most  general  value 
of  the  integral  belonging  to  the  exponent  k  contains  k  -{-  i  arbitrary 
constants.  Let  us  suppose  (which  we  may  do  without  any  loss  of 
generality)  that  when  the  arbitrary  function  C  {r),  which  enters  as  a 
factor  in  all  the  coefficients  of  the  series  2g^{r)x''+'' ,  becomes  unity, 
the  function  g{x,  r)  becomes  h{x,  r)  ;  then  g{x,  r)  =  C{r)h{x,  r),  and 
consequently 

(40)  g'{x,  r)  =  C/i^{x,  r)  +  kC7i^-^{x,  r)  +  .  .  .  +  C,/i{x,  r), 

where  g"^,  C'^,  h"^  denote  derivatives  of  g,  C,  h  with  respect  to  r. 
The  functions 

h\x,rk),     h^-'{x,rk),     .  .  .  ,     h{x,rk) 

are  linearly  independent,  and  consequently  the  integral  g^{^,  ^k) 
contains  the  k -{-  i  arbitrary  constants  C,  C,  .  .  .  ,  C^ .  We  have 
thus  formed  the  general  integral  belonging  to  the  exponent  k  of  the 
differential  equation  without  the  aid  of  the  integrals 


I04  LINEAR  DIFFERENTIAL   EQUATIONS. 

In  the  particular  case  of  the  root  r^  the  integral  ^(;r,  r^  contains 
only  one  arbitrary  constant.  It  follows  at  once  from  what  precedes 
that  one  integral  of  the  differential  equation  P{y)  =  O  is,  to  an  arbi- 
rary  constant  pres,  completely  determined  by  the  following  condi- 
tion:  If  the  integral  is  divided  by  x''  the  quotient  must  be  uniform, 
and  must  also  be  finite  for  ;ir  =  o,  r  denoting  a  root  of  the  algebraic 
equation  fi^)  =  o  which  does  not  exceed  any  other  root  of  this 
equation  by  a  positive  integer. 

We  have  above  made  an  assumption  concerning  the  form  of  the 
arbitrary  function  ^(r),  in  order  to  prevent  the  coefficients  of  the 
series ^(;r,  r\  from  becoming  infinite:  viz.,  we  have  assumed 

g{f)  =f{r  +  i)f{r  +  2)  .  .  .  /(r  +  e)C{r) ; 

where  C{r)  is  an  arbitrary  function  of  r,  and  where  e  denotes  the 
greatest  (integral)  difference  between  two  roots  of  any  group  of  roots 
o(  f{r)  =  O.  It  is  obvious  that  e  may  take  any  other  value  greater 
than  this,  and  the  foregoing  results  will  still  hold.  The  value  of  this 
liberty  of  choice  of  e  is  seen  if  we  wish  to  actually  compute  the  value 
of  the  series  ^(^,  r)  up  to  a  given  power  of  x,  say  x''+^.  In  this 
case  assume  e  =  k;  then  all  of  the  assumed  coefficients ^i,(r)  will  be 
integral  functions  of  r,  and  so  the  inconvenience  of  differentiating 
fractional  forms  will  be  evaded.  Of  course  it  is  understood  that  the 
arbitrary  function  C{r)  must  be  so  chosen  as  not  to  become  infinite 
for  any  of  the  considered  values  of  r.  For  further  remarks  on  this 
point  the  reader  is  referred  to  Frobenius's  memoir. 

We  have  seen  both  by  Fuchs's  method  and  that  of  Frobenius  that 
logarithms  generally  appear  in  the  integrals  of  2l  group.  ^\\&  groups 
of  integrals  arise,  it  is  to  be  remembered,  from  the  fact  that  certain 
roots  of  the  indicial  equation  differ  from  each  other  by  integers,  in- 
cluding zero.  The  separation  of  these  roots  into  groups,  as  already 
described,  gives  rise  to  the  corresponding  groups  of  integrals.  It  may 
be,  however,  that  logarithms  will  not  appear  in  a  group  of  integrals. 
Fuchs  investigated  the  conditions  necessary  to  be  satisfied  in  order 
that  no  logarithms  should  appear  in  a  given  group.  The  reader  is 
referred  to  Fuchs's  memoir  in  Crelle  (vol.  68)  for  his  investigation  of 
this  subject,  and  also  to  one  by  Cayley  in  Crelle,  vol.  lOO.  The  test 
to  be  applied  in  order  to  ascertain  whether  or  not  logarithms  exist  is 


FKOBENIUS'S  METHOD.  I05 

obtained  by  Frobenius  in  a  very  simple  manner  in  his  memoir  which 
we  are  now  considering.  Before  taking  up  Frobenius's  remarks  on 
this  subject,  we  can  show  (Tannery,  p.  167)  that  if  in  a  group  two 
roots  are  equal,  say  r^=^  r^,  then  logarithms  necessarily  appear  in 
this  group.  The  integrals  belonging  to  r^  and  r^  are  in  general  of 
the  form  (in  the  region  of  ;tr  =  o) 

where  the  functions  (p  are  uniform  and  continuous  in  the  region  of 
^  =  O.     If  now  in  P{j')  =  o  we  make  the  transformation 

we  know  that  the  integral  z^  of  the  linear  differential  equation  of 
order  u  —  i  in  s'  belongs  to  the  exponent  r„  —  r,  —  i  ;  but  since 
r^  :z=  r^ ,  we  have  in  this  case  r^  —  r,  —  i  ^  —  i  as  the  exponent  to 
which  s^  belongs,  and  therefore  ^,  is  of  the  form 

where  the  constant  y^  does  not  vanish,  and  where  0[x)  is,  in  the  region 
of  ^  =  O,  a  uniform  and  continuous  function.     From  this  we  see  that 

is  of  the  form 

A  logx-\-  ip{x), 

and  consequently  _;;/,  is  of  the  form 

It  is  clear  from  this  simple  illustration  that  if  there  are  equal  roots 
in  a  group,  logarithms  must  necessarily  appear  in  the  corresponding 
group  of  integrals. 

By  aid  of  equation  (37)  the  Frobenius  method  leads  us  to  the 
same  conclusion  ;  we  have  seen,  in  fact,  that  in  the  integrals  belonging 
to  an  a-.tuple  root  o{  /{r)  =  o  there  are,  in  general,  at  least  a  —  i 
logarithms,  and  so  in  order  that  no  logarithms  exist  in  the  group  we 


I06  LINEAR  DIFFERENTIAL   EQUATIONS. 

must  have  the  roots  r^,r^,  .  .  .  ,  r^,  among  which  the  r  of  equa- 
tion (37)  is  to  be  found,  all  different.  We  can  now  quickly  find  by 
Frobenius's  method  the  conditions  to  be  satisfied  in  order  that 
logarithms  may  not  appear  in  a  given  group  of  integrals.  We  have 
already  seen  that  the  series  ^'^(;ir,  r^)  of  equation  {^y)  may  represent, 
without  the  aid  of  the  series  ^'^-'(;ir,  r^_  I ),  .  .  .  ,  ^(jr,  r„),  the  most 
general  integral  of  the  differential  equation  P[y)^  o  belonging  to 
the  exponent  r^ .  It  follows  now  at  once  that  in  order  that  the  in- 
tegral g^{x,  Tk)  may  contain  no  logarithms  we  must  have  all  of  the 
functions  ^^(r)  zero  of  order  ^  for  r  =  r^.     Now, 

g{r)h^{r) 
(4^)  S^ (')  =  /(r-f  i)/(r+2)  .  .  ./(r+r)  = 

in  which  gir)  is  zero  of  order  /'  for  r  =  r^.     The  function 

(42)  H^{r),     _  ^^^  ^  ^^^^^  +  2)  .  .  .  /(r  +  T')  ' 

must,  therefore,  not  become  infinite  for  r  =  r^.  Since  the  functions 
ga{^)  are  connected  by  the  formula  (see  equations  (9) ) 

(43)  <^./(^  +  ^)  +<r.- 1/.  (^  +  ^  -  0  +  •  •  •  ^^f"  W  =  o, 
and  since 

(44)  ^"^'''''"^y 

we  have  for  the  functions  //the  formula 

(45)     If.Ar+r)-\-I/^_,/^{r  +  v -!)-{-  .  .  .  +/^/,(r)  =  o. 

If  now  //;,  _  I ,  Hy  _  2 ,  .  .  ,  //  are  all  finite  for  r  =  rj^ ,  the  same  must 
also  be  true  for  Hvf{r  -\-  v). 

Now  from  (44)  we  have  //=  i,  and  consequently  Hv{r^  will  be 
finite  for  all  values  of  v,  provided  only  it  does  not  become  infinite 
for  such  values  of  v  as  make  rk-\-  y  "s.  root  of  the  equation  f{r)  =  o ; 
that  is,  for  values  of  v  equal  to  some  one  of  the  quantities 


FROBENIUSS  METHOD. 


lOT" 


It  is  easy  to  see  now  that  in  order  that  Hv  {r^  shall  be  finite  for 
r  =  ri:_i  —  r^  it  is  necessary  and  sufficient  that  hv{r)  shall  be  zero 
of  the  first  order  for  r  =  r^ .  For  v  =  Tk-^  —rk  and  r  =  rk  we 
have  now 

(45)     H,{r)f(r+  v)  =^^^^  ,)/(-  +  ^t-'-  •/(-+--■) 

and  K{r)  is  zero  of  order  one.  In  order  that  H^{r)  shall  be  finite  it 
is  now  necessary  that  hj  (r^)  shall  vanish.  Continuing  this  process, 
we  readily  find  the  following  conditions  which  must  be  satisfied  in 
order  that  the  integral  belonging  to  the  root  r^  of  the  indicial  equation 
f{r)  =  o  may  contain  no  logarithms.  Defining  the  function  ky  (r) 
by  the  equation 


(46)     {-iyAy{r)  = 
j/(r+r-  I),    /^{r^v-2), 

j/(r+  y  _  i),    /^(r-^  V-  2), 

o  ,    /(r  +  r  -  2), 


/._.(r+i),    /.(r) 
/._,(r  +  i),    /,-^{r) 
/,_3(r+  I),    f,-.{r) 


O  .  O  ,  .  .  .  ,  /(r+i),     /(r) 

then  for  r  ^  rk  the  following  equations  must  be  satisfied  : 
h,.{r)  =  o  for         r  =  r^_^_r^, 

K  (^)  =  o  for        r  =  r^  _^  —  r^  , 


(47) 


VfK-\r)=  o 


for 


^^ 


CHAPTER   V. 

LINEAR   DIFFERENTIAL   EQUATIONS   ALL   OF   WHOSE   INTEGRALS 

ARE   REGULAR. 

In  what  follows  the  critical  point  under  consideration  will  be 
taken,  as  in  the  preceding  chapters,  as  4:  =  o.  The  results  obtained 
for  this  point  can  be  changed  into  the  corresponding  results  for  any- 
other  critical  point  x^  by  changing  x  into  x  —  x^\  the  functions  0,y 
or  [M,  N)  will  of  course  be  different  for  each  critical  point,  but  will 
always  be  uniform.     It  has  been  seen  that  functions  of  the  form 

(i)      F  =  ,r'[0„  +  0jog  ^  +  ....  +  0,  log*^], 

•or,  for  a  critical  point  x  =  x^, 

F={x-  ,tv)'-  [0„  +  (P,  \og{x  _  ;r,)  -f  .  .  .  +  0,  log*  {X-X;)\ 

play  an  important  part  in  the  theory.  Since  the  uniform  functions  0 
contain  (for  the  present)  only  a  finite  number  of  negative  values  of 
^,  or  X  —  Xi,  and  since  for/  positive  the  products 

x''  log''  X,     {x  —  xy  log*  {x  —  x^ 

are  zero  for  x  =  o  and  x  =^  Xi  respectively,  it  follows  that  we  can 
•always  find  a  number  r  such  that  the  product 

x~  ''F    or     [x  —  x^  '/^ 

shall  be  different  from  zero,  and  shall  become  infinite  in  the  same 
way  as  the  function 

(2)  a  -{-  ft\ogx  ^  .  .   .^\  Xog'x, 

or  «'  +  /^  log  {x-xi)  4-  ...  4-  A  log*(;ir  —  x^, 

"where  a,  (3,  .  .  .  ,  X  are  constants. 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     IO9 


We  will  use  now  only  the  critical  point  ;f  =  O,  as  the  necessary- 
changes  for  any  other  point  are  obvious. 

The  function  F  oi  {\)  is  now  said  to  belong  to  the  exponent  r. 

If  two  functions  /^and  F^  belong  to  exponents  r  and  r^  respec- 
tively, the  product  FF^  will  belong  to  the  exponent  r  -{-r^;  also, 


SFF^  =  SFX  SF,,     S 


'F' 
-F- 


SF 
=  -pr?^  ,  etc. 
SF, 

dF 


If  F  belongs  to  the  exponent  r,  then  in  general  —  will  belong  to 

the  exponent  r  —  i.  A  case  of  exception  arises,  however,  when, 
r  ^  o,  and  when  at  the  same  time  F  does  not  become  infinite  for 
;f  =  o.     Suppose 

F  ■=,  x'\(f),  +  0  Jog  ;ir  +  .  .   .  +  0*  log*  x\, 

r  being  the  exponent  above  described,  it  follows  that  the  uniform 
functions  0  contain  only  positive  integer  powers  of  x  in  their  devel- 
opment, and  are  not  all  zero  for  x  =  o.     We  have  now 


dF 
dx 


^x" 


d4>j. 


log^,r  + 


+  ^^ 


rcpk  +  ^  - 


d<f>k 


dx 


log*  X, 


The  coeflficients  of  the  different  powers  of  log,r  cannot  all  vanish  for 
;r  =  o ;  for  if  they  could  we  should  have  the  system  of  equations 


'  ^(Pp  +  (/  +  i)0/  +  i  =  O, 


(3) 


^0* 


but  this  system  gives  us 


=  o; 


=:  O 


for  X  =  o,  which  is  contrary  to  hypothesis.     If,  however,  r  =  O,  it  is 

only  necessary,  in  order  that  all  the  coefficients  of  log .;»;  vanish  in 

dF 

— ,  i.e.,  that  equations  (3)  be  satisfied,  that  we  have 


0* 


<l>k 


110 

for  X  =  o 

value  of  00,  and 


LINEAR  DIFFERENTIAL   EQUATIONS. 


In  this  case  F  reduces  to  the  finite  and  non-vanishing 
dF 


dx 


belongs  to  the  exponent  zero,  or,  in  certain 


cases,  to  a  positive  integral  exponent. 

It  is  easy  to  see,  if  F  belongs  to  the  exponent  r,  that  by  properly 
choosing  the  constant  of  integration  the  function 

fFdx 

will  be  of  the  same  character  as  F,  and  will  belong  to  the  exponent 
r-\-\.  The  preceding  results  (taken  from  Tannery's  memoir)  may 
be  briefly  expressed  as  follows  : 

1.  Every  regular  function 

F  =  x'lspk  log*^  +  0A    I  log* " '  ^  +  .  .  .  +  0o] 

has  a  derivative 

(  r            • ,                                    k(bk  log* "  ^x 
(4)  x^  \  -  [0.1og*^+  .  .  .  +  0„]  +  -^^^ 

which  is  also  regular. 

2.  Every  product  or  quotient  of  regular  functions  is  also  regular. 
The  following  derivation  of  the  form  of  the  differential  equation  all 
of  whose  integrals  are  regular  in  the  region  of  a  critical  point,  is  taken 
from  Jordan.  Denote  by  _>/.,...  ,^„  a  system  of  regular  inte- 
grals of  the  equation  in  the  region  of  the  critical  point  x  =  o,  and 

let  ji"  denote  the  differential  coefficient  -^ ;  the  equation  can  be 

written  in  the  form 


(5) 


y\  y: 
y,  y: 


.    yn 
,    yJ 

r" 


=  o. 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     Ill 

The  coefficients  of  y,  y' ,  -  -  -  ,  j"  are  the  principal  minors  of  the 
determinant  corresponding  in  order  to  these  quantities,  and  they  are 
obviously  sums  of  regular  functions,  such  as 

(6)  x\iPi\og'x  +  ...]+  ^'•i[^,/  log'-.^  +  ...]  + 

Now,  when  x  turns  round  the  critical  point  ;r  =  o,  the  integrals 
I 

y,,    .  .  .  ,   /«, 

and  their  derivatives, 
I  J,",    .  .  .  ,    y»^     a=  I,  2,  .  .  .  ,  n, 

submit  to  the  same  linear   substitution ;  the  coefficients  of  y,  y', 
.  .  .  ,  y"  therefore  reproduce    themselves  each  multiplied    by   the 
determinant,  say  d,  of  the  substitution.     [If  we  divide  the  equation 
through  by  the  coefficient  oi  y"  ,  the  coefficients,  say /j ,  A  >  •  •  •  >  A 
oi  y' ,  y" ,  .  .   .  ,  jj/"  ,  will  therefore  reproduce  themselves  exactly  after 
the  substitution  ;  in  other  words,  they  are,  as  by  the  original  hypothe- 
sis, uniform  functions.]     In  order  that  an  expression  of  the  form  (6) 
;  may  reproduce  itself  multiplied  by  the  determinant  S  after  x  turns 
I  round  jt-  =  o,  it  is  clear  that  the  logarithms  must  all  disappear,  and 
I  that  the  exponents  r,  r^,  .  .  .  can  only  differ  by  integers  from  the 
[  quantity 
i  logc^ 


27rt 


=  ,  say,  /?. 


The  coefficients  of  the  differential  equation  (5)  are  therefore  of 
the  form  x^P,  where  P,  like  the  functions  0,  is  a  uniform  function 
of  X,  having  x  =0  only  as  an  ordinary  point  or  a  pole. 

It  follows  also  from  the  remark  in  brackets  that  A,  A'  •  •  •  >A 
can  have  x  =  o  only  as  an  ordinary  point  or  a  pole. 

Fuchs's  theorem  regarding  equations  which  have  a  system  of 
independent  integrals  all  of  which  are  regular  may  be  stated  as 
follows : 

In  order  that  the  linear  differential  equation 

d"y  d"  "  'j 


112  LINEAR  DIFFERENTIAL  EQUATIONS. 

may  have  a  system  of  linearly  independent  regular  integrals  in  the 
region  of  the  point  x  =  o,  it  is  necessary  and  sufficient  that  each  co- 
efficient, say  pi ,  of  the  equation  shall  have  the  point  x  =  o  for  an 
ordinary  point  or  a  pole  ;  in  case  this  point  is  a  pole,  its  order  of  mul- 
tiplicity must  not  be  greater  than  i. 

The  first  part  of  this  theorem  has  already  been  established.  The 
groups  of  integrals  which  have  been  formed  corresponding  to  the  dis- 
tinct roots  s^,  s^,  .  .  .  of  the  characteristic  equation  have  belonging 
to  each  of  them  a  certain  exponent  r  defined  by 

log  Sj  _ 

27ti      ~^'' 

the  values  r,,  r^,  .  .  .  derived  from  ^, ,  ^^ ,  .  .  .  ,  and  associated  with 
each  group  of  integrals,  must  then  differ  from  each  other  by  quanti- 
ties other  than  integers.  As  a  change  of  r,-  into  r^-  -f- ;;/,  where  ;//  is 
an  integer,  would  leave  Si  unchanged,  it  is  clear  that  the  factor  ;f^•  in 
the  group  associated  with  Si  might  have  different  values  of  r,-  in  the 
different  integrals  of  the  group,  provided  those  values  differed  only 
by  integers.  In  establishing  the  second  part  of  Fuchs's  theorem  we 
shall,  however,  arrive  at  an  algebraic  equation  (the  equation /(r)  =o 
of  the  last  chapter)  determining  the  exponent  r  for  each  group  with- 
out ambiguity.  Jordan  proves  this  second  part  of  the  theorem  in 
a  very  brief  and  elegant  manner.  Fuchs's  proof,  which  is  rather 
longer,  will  be  given  later.     If  we  transform  the  equation 

dy        d"-y  , 

by  the  substitution 

y  =  ^fh 

where  7;  is  the  new  dependent  variable,  and  X  is  of  the  form 


x"  2  Ci  X' 


i  =  o 


and  therefore 


(7)  —=x-^2eiX\ 


EQUATIONS  ALL   OF    WHOSE  INTEGRALS  ARE  REGULAR.      II3 

we  have  the  equation 

d^n  d^  ~  'w 

(8)  x-£,+,:^,+  ...+,:n  =  o. 

The  values  oi  q^' ,  .  .  .  ,  qj  are,  from  (12),  Chap.  Ill, 


,       n{n  —  \)d'X 


dX 


1.2       dx 


^"  +  ^^>^+^^^ 


Qn 


d"X    .        d'^-'X 


dx 


n'   +  A  -J-^--   +     •     •     •     -\-pnX 


or,  dividing  J3y  X,  calling  the  new  coefficients  q^,  g^,  .  .  .  ,  g„,  and 

d'X 

writing  X*'^  for  —7-^  , 

'^  dx'  ' 

X' 


I 


(9) 


n{n-\)X"  X' 

q.  =  ~^-r"x  +  ^A  ^  +  A, 


X'  .  X" 

Now  -p  has  obviously  the  point  ;r  =  o  as  a  pole  of  order  i  ;  —-  has 


XW 


;f  =  o  as  a  pole  of  order  2  ;  and  in  general  -y  has  x  =  0  as  a  pole 
of  order  /. 

If  then  A»A»  •  •  •  ypn  have  ;f  =  o  as  a  pole  of  at  most  the 
order  i,  2,  .  .  .  ,  n,  respectively,  it  follows  that  the  coefficients  ^,, 
^, ,  .  .  .  ,  ^„  in  the  equation 


(10) 


d^l^  d^-^-,1 


114  LINEAR  DIFFERENTIAL   EQUATIONS. 

have  ;r  =  o  as  a  pole  of  at  most  the  orders  1,2,...  ,  n,  respectively. 
(It  is  easy  to  see  that  the  order  of  multiplicity  of  the  pole  x  ■=^  o  \x\ 
qi  may  be  less  than  i,  but  under  the  hypothesis  as  to  the  coefficients 
/,■  it  can  never  be  greater  than  i). 

If  now  the  equation  my  with  coefificients  /  has  all  of  its  integrals 
regular  in  the  region  of  the  critical  point  ;r  =  o,  it  follows,  from  the 
relation 

J  /  ~  00 

that  all  the  integrals  of  the  transformed  equation  (10)  are  regular; 
conversely,  if  all  the  integrals  of  (10)  are  regular  and  its  coefficients 
possess  the  property  in  question,  then  all  the  integrals  of 

will  be  regular,  and  the  coefflcients/  will  possess  the  same  property, 
since  (11)  is  derived  from  (10)  by  the  transformation 

We  have  seen  that  the  equation  (11)  with  uniform  coefficients 
always  admits  of  at  least  one  integral  of  the  form 

where  0„  is  a  uniform  function  containing  only  positive  powers  of  x, 
and  different  from  zero  when  x  =  o.  If  we  replace  the  function  X 
byjj/„,  =  ^'■00  >  equation  (10)  retains  its  form,  but  the  coefficient  q„ 
becomes 

which  vanishes,  and  therefore  (10)  must  be  written  in  the  form 


I 


I 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE   REGULAR.     II5 


■a  differential  equation  of  the  «th  order  having  i,  or  any  constant, 
as  an  integral ;  such  an  integral  is  of  course  regular. 
Consider  the  differential  equation  of  the  first  order 


<I3) 


1  +  ^-^  =  - 


which  admits  a  regular  integral, 


The  quotient. 


I  dy 
y  dx'' 


admits  obviously  the  point  ;ir  =  o  as  a  pole  of  order  i  at  most,  and 
therefore  P  must  have  ;r  =  o  for  a  pole  of  order  i  at  most ;  for  if  P 
had  ;ir  =  o  for  a  pole  of  order,  say,  or  >  i,  then  there  would  be  terms 

in  P  which  could  not  cancel  with  any  terms  in  —  —  . 

y  dx 

Again,  take  the  differential  equation  of  the  second  order 


(14) 


d'y 


dy 


2?  +  ^ji  +  e/  =  o, 


having  all  its  integrals  regular  in  the  region  of  the  point  x  =0.    Sup^ 
pose  F,    =  x''(p^ ,  to  be  one  of  them  ;  then  writing 

y  =   V3, 

it  is  clear  that  s  must  be  regular;  substituting,  we  have 


dx' 


+ 


2dV 

Lvd^ 


-hP 


dz  I 

Ix^  Y 


""+-'^r+er 


_dx: 


dx 


O, 


dz 


■or,  making  ~p  =  v  {v  is  therefore  regular), 
dv     .    V  2dY 


(15) 


dv        r 


Y  dx 


^-P 


V  =  o. 


ii6 


LINEAR  DIFFERENTIAL   EQUATIONS. 


As  already  seen,  the  coefficient  of  v  here  has  ^  =  o  as  a  pole  of  order 

I  dV 
I  at  most,  and  since -.^^  — -  possesses  this  property,  it  follows  that  P 
V   ax 

has  ;ir  =  o  as  a  pole  of  order  i  at  most. 

"Write 


w  ^^  e  ^fPd^  ,     y 
then  substituting  in  (14),  we  have 


wv\ 


(16) 


d'v 
Ix 


-i  +  /^^ 


o, 


2   dx        4 


If  jj/,  and  j^  are  two  independent  regular  integrals  of  the  given  equa- 
tion, and  corresponding  to  them  are  v^  and  v^ ,  then 

(17)  z\  ^  y .e^-^P^'^ ,     7',  =  j/i^'^'^-^. 

From  what  has  been  found  concerning  P  it  follows  that 

is  regular,  and  therefore  that  z\  and  v^  are  regular.     The  equation 


(18) 


d^v 


dx 


,  +  /^ 


has  then  its  integrals  regular,  and  consequently 

I  d\^ 
V  dx'' 

admits  ;t-  =  o  as  a  pole  of  order  2  at  most,  and  consequently  /,  and 
therefore  Q,  admits  ;ir=  o  as  a  pole  of  order  2  at  most.  (It  is  easy 
to  see  from  (18)  that  P  and  Q  have  the  required  property.)  A 
similar  process  might  be  employed  for  the  differential  equation  of 
the  third  order,  but  it  is  rather  long,  and  besides  unnecessary.  These 
two  illustrations  show  the  truth  of  the  theorem  for  n=  i,  2. 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE   REGULAR.     11/ 

Returning  now  to  equation  (12),  write  -^  =  //,  and  the  equation 
becomes 

(.9)  s^^+?.s^  +  ---  +  ?"--''  =  °- 

The  integrals  of  this  are  the  derivatives  of  the  integrals  of  (12);  but 
these  latter  are  regular,  therefore  the  integrals  of  (19)  are  regular. 
Suppose  now  the  theorem  to  hold  true  for  (19);  then  ^,-  admits  ^  =  0 
as  a  pole  of  order  i  at  most.  The  theorem  is  then  true  for  equation 
(12)  ;  and  since  (12)  is  derived  from  (11),  by  the  transformation 


(20)  -n  =  -^y 

it  is  also  true  for  (11).     We  have  then  for  the  region  of  j^  =  o  the 
equations 

P  P  P 

(21)  A  =  — .     A  =  — i.   •  •  •  .    A=  -;;. 

Avhere  P,,   P^,   .  .  .  ,   P„  are  in  the  region  of  ;r  =  o  uniform  con- 
tinuous functions,  such  as 


I 


(22) 


^.    =    «o   +   ^1-^  +   ^.'^''   + 

P,  =  d,  +  b,x  +  Kx-  + 


If  we  know  the  radius  of  the  common  circle  of  convergence  of 
these  series,  say  p,  then  the  values  of  the  coefificients  a,  b,  .  .  .  will 
be  limited  by  the  inequalities 

mod.  a,„  ^  — ,     mod.  ^^n  ^  -^ ,  ■  •  ■ 

where  Mis  a.  properly  chosen  constant.     If  we  substitute  in  (11)  for 
y  the  value 

23)  7  =  x'-, 


Il8  LINEAR  DIFFERENTIAL   EQUATIONS. 

we  have  as  the  result 

(24)  F{r)x''  +  0,(O-^''  +  '  +  0,(r>''  +  ^  +  .  .  .  ,     =  x'-^ix,  r\ 
where 

(25)  F{r)  =  r{r  -  i)  .  .  .  (r  -  «  +  i)  +  ^oK^—  l)  •  •  •  (r  -  «  +  2) 

+  .  .  .  +  <^„r(r  _  I)  .  .  .  (r  -  «  +  3)  +  .  .  .  „ 

(26)  0,„(r)  =  «,„r(r  —  I)  .  .  .  (r  —  ;/  +  2) 

+  b,„r{r  -  i)  .  .  .  (r  -w+  3)  +  .  .  . 

The  equation  F{r)  =  o  is  called  by  Fuchs  the  "■  determinirende 
Fiindamentalgleichungr  following  Cayley  {Quart.  Jotirn.  of  Math., 
1886)  we  will  call  it  the  Indicial  Equation.. 

We  will  now  give  Fuchs's  investigation  concerning  the  forms  of 
the  coefficients  in  the  case  where  the  differential  equation  has  all  of 
its  integrals  regular.  Let  o,  x^,  x^,  .  .  .  ,  x^  denote  the  critical 
points  of  the  integrals,  and  write 

ip{x)  =  x{x  —  x^  .  .  .  {x  —  x^ ; 
then  the  differential  equation  under  consideration  has  the  form 

.,.x         dy_       P^d_-^        PJ^d_;-^y 
^"^^         dx''^    tfix)      dx'     "^    f{x)dx"-' 

I   P.p(-y) 
+  •  •     ^  r{x)  ^ 

where  P^p  denotes  a  polynomial  in  x  of  the  degree  kp,  or  of  a  lower 
degree. 

We  have  seen  in  Chapter  III  that  a  system  of  fundamental  inte- 
grals of  the  equation 

can  be  obtained  of  the  form 

(29)  f.^    7.=  J, /-A-    y.=  y.f^.dxft.dx,  .  .  . 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.      II9 

where  z^,  /, ,  .  .  .  are  integrals  of  linear  differential  equations  of 
orders  n  —  i,  n  —  2,  .  .  .  respectively.  It  is  obvious  that  these 
auxiliary  functions  can  be  so  chosen  that  the  fundamental  integrals 
J, ,  .  .  ,  ,  y„  shall  submit  to  the  substitution  (80)  (Chapter  III)  when 
the  variable  turns  round  the  point  jir  =  o.  The  integrals  of  the 
equation  of  the  71^^  order  are 

y,,     y.,     •   •  •   >     Jnl 

those  of  the  equation  of  order  n  —  i  are 

""i  >       -^a  >       •    •    •    >       ■^n  -  I  > 

those  of  the  equation  of  order  n  —  2  are 

^1 )     ^2  >     •  •  •  >     '»  -  2 ) 

etc.  It  is  easy  to  see  that  the  integrals  j/, ,  ^-j ,  /, ,  .  .  .  are  all  of  the 
form  x'(p{x),  where  0(^)  is  a  uniform  function  different  from  zero  for 
X  =  0,  and  that  they  belong  respectively  to  the  exponents 

^1,     ^  —  ^  —  I.     r,  —  r,~i... 

The  case  where  the  difference  of  two  exponents,  say  r^-  and  r^ ,  is 
zero  gives  rise  to  the  "  case  of  exception  "  mentioned  above  ;  the 
difificulty  arising  here  can,  however,  be  removed  by  replacing  one  of 
the  two  corresponding  integrals  by  a  proper  linear  function  of  both. 
For  example,  take  the  differential  equation 

^'  _      ^  —  4     _  ^   I       -y  —  3         _ 
dx''       2x{x  —  2)       dx    '   2x''{x  —  2y  ~     ' 

this  admits  as  solutions  the  two  independent  functions 

x^       and       {x'^  —  2xy  , 

which  both  belong,  in  the  region  of  x  =  o,  to  the  exponent  ^ ;  we 
can,  however,  replace  them  by  the  independent  functions 

Xi  ,       {x"  —  2xf    —   V  —  2  .  Xi  , 

or 

xi 

^^'       {X  -  2y  +  ^^^2    .    Xi  ' 


I20  LINEAR  DIFFERENTIAL   EQUATIONS. 

belonging  to  the  exponents  \  and  f  respectively.     This  "  case  of  ex- 
ception "  being  borne  in  mind,  we  need  not  refer  to  it  again  unless 
it  is  absolutely  necessary  in  some  particular  case. 
Suppose  now  that  y^  is  known  and  is  of  the  form 

(30)  y,  =  x^^(p{x), 

and  satisfying  from  (80')  the  relation 

^}\  ^  s,y,\ 
suppose  further  that  we  write,  as  we  may, 

(31)  y.^yj-4x\ 

where  Sy^  =  s^y^^  s^y^,  and  where  j,  belongs  to  the  exponent  r,  r 
we  have 

d  y^ 

(32)  ^'"^^7/ 

from  which  follows 

(33)  -^1  —  dx      jKi  ~  dx        s^y^  dx  y^ ' 

From  this  it  is  clear  that  z^  is  a  uniform  function  belonging  to  the 
exponent  r^  —  r^  —  \.  It  is  easy  to  show  that  the  remaining  in- 
tegrals ^j ,  .  .  .  ,  ^„-i,  which  with  z^  form  a  fundamental  system  for 
the  equation  of  the  {n  —  i)'^  order  belong  to  the  exponents 

r,  —  r,-\,     r,—  r,-  I,     .  .  .  ; 
and  also  that  in 

y.  =  yj^.dxft.dx 

the  function  /,  is  uniform  and  belongs  to  the  exponent  r,  —  r^—  i. 

By  continuing  in  the  manner  indicated,  we  find  finally  that  the 
functions 

/, ,    J,^. .    Ji^/i '     •  •  •  '    7i^.^.  •  •  •  «^i 

belong  respectively  to  the  exponents 

r,,     r,  -  I,     r,-2,     .  .  .  ,     r^-{n-  i). 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     121 
Prom  equation  (i6),  Chapter  III,  we  have, 

(34)  .  D  =  Cy:^zr'  .  .  .  w,, 

where  C  is  a  non-vanishing  constant.  It  follows  then  on  substituting 
for  J/,  its  value,  and  remembering  that  z^,  t^,  .  .  .  ,  w^  are  all  uni- 
form functions,  that 

(35)  Z>  =  ;r'-i  +  '-»+- ••+'■«-— r"-V(;ir); 

where  ^(;r)  is  uniform  and  continuous  in  the  region  x  =  o,  and 
does  not  vanish  when  ^  —  o.  A  similar  form  is  at  once  obtained 
for  the  point  infinity :  it  is  only  necessary  to  change  the  variable  by 
the  relation 

I 

.and,  as  before,  investigate  the  point  /  =  o.     We  shall  return  to  this 
point  in  a  moment.     We  have 

A 

(36)  A  =  -  ^ ; 

denote,  as  above,  by  6  the  determinant  of  the  substitution  arising 
from  travelling  round  x  =  o,  and  write 

log  d 

It  follows  now  at  once  that 

n  =  x"'+^ip{x), 


(38) 

'  I),  =  x"''+^7p\x); 

m  and  in'  being  integers,  and  ip{x),  ^X-^)  uniform  and  continuous 
functions  of  x  in  the  region  of  x  ^  o.  In  developing  these  deter- 
minants no  logarithms  will  appear.  These  results  have  already  been 
obtained.  Suppose  now  we  multiply  each  element  in  Di  and  D  by 
X  raised  to  the  power  denoted  by  the  negative  of  the  exponent  to 
which  the  corresponding  element  belongs.  The  determinant  D^  will 
then  be  multiplied  by 

(  "  n  {n  -  i)        .  ) 


122  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  D  by 

The  quotient, 


'^n 


n{n  —  \) 


\. 


'BD  ' 

Di     .  I 

differs  from  -^  ,  i.e.,  —pi,  by  the  factor  -,-.     It  follows  now  at  once 

that/,-r'  is  a  uniform  and  continuous  function  of  x  which  may  be- 
come zero  for  ^  =  o,  but  may  not  become  infinite  for  this  value  of  x. 
We  have  then,  finally. 


(39) 


A  =  -; 


where  /*,■  is  a  uniform  and  continuous  function  in  the  region  of  ;ir  =  o  p 
and  if  the  integrals  are  regular  in  the  regions  of  all  the  critical  points 
.  ,  Xp  ,  then 


o,  X,, 

(40) 


A  =  S 


where  ?/;  :=  x{x  -^  x^  .  .  .  (x  —  Xp),  and  /*,  now  denotes  a  uniform 
and  continuous  function  in  all  the  plane.  The  equation  has  now 
the  form 

To  study  the  point  co,  write  x  =  — .    Transforming  the  equation, 
we  find  without  difficulty 


(42)     ■^^  +  ««,„-,^ 


dt 


d"-'y  I 


—  a 


n  -  i.n  -  2 


51 

tp  f 


+  -'- 


dr 


+  .  .  .  =0;, 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE  REGULAR.      1 25 


where  -^  =  7  has  to  be  substituted  in  y-,  /*j ,  .  .  .  ,  /*„ .  The  quan- 
tities ^4, 4-1 ,  «*, 4.-2 )  •  •  •  are  integers,  the  first  of  which  =^  k{k  —  i). 
The  point  /  =  o  is  now  a  pole  (or  ordinary  point)  of  order  at  most 
=  1,2,  .  .  .  ,  n,    for  each  of  the  coefificients  of 


dr^  '         dt"  -  ^ 


y 


be  developable  in  positive 


it  is  therefore  necessary  that  --  ,  -| ,  . 

ascending  powers  of  t,  and  that  the  first  term  in  each  of  these  de- 
velopments shall  be  of  the  degree  1,2,  .  ,  .   ,  ii,  respectively.     Now,, 


Mf--^' 


7-"' 


/P+x4_^^/P_|_.     .     .. 


we  must  therefore  have 


(43) 


-  H 4-  .  .  .   =  djp  +  dJ<^  -  '  +  . 


tP 


tP 


-  ^2p  I        pp  -    I  I  1  I  2  I 


That  is,  P, ,  /'j ,  .  .  ,  Z*,,  are  polynomials  of  the  degrees 


P,     2p,     3p, 


7ip  ; 


where  p  -j-  i  is  the  number  of  finite  critical  points. 

We  shall  now  give  Fuchs's  proof  (which  has  only  so  far  been 
faintly  indicated)  of  the  converse  of  the  theorem  just  proved  ;  viz.,,. 
we  shall  prove  that  every  linear  differential  equation  of  the  form 


(44)  2i^  + 


dy     M^d"-y     M.^d"-y 


X  dx 


;r4:  + 


x""  dx' 


-.+     • 


+  -^ry  =  o, 


where  M^ ,  M^,  .  .  .  ,  M„  are  uniform  and  contimtous  finctions  of  x 
in  the  region  of  tJie  critical  point  x  =  o^  admits  in  the  region  of  this 
J>oint  a  system  of  fundamental  integrals  all  of  zvhich  are  regular. 


124 


LINEAR   DIFFERENTIAL   EQUATIONS. 


The  method  employed  in  Fuchs's  proof  is  the  same  as  that  used 
by  Weierstrass  in  his  general  proof  of  the  existence  of  an  integral  of 
an  algebraic  differential  equation.  Instead,  however,  of  referring 
directly  to  Fuchs's  memoir,  we  shall  follow  Tannery's  exposition  of 
the  same. 

From  what  has  been  shown  we  know  that  (44)  admits  at  least 
one  integral  of  the  form 


(45) 


y  =  x''(p{x), 


where  (j){x)  is  a  uniform  and   continuous  function  of  x,  and  is  not 
zero  for  x  ^  o.     If  now  we  substitute 


(46) 

in  (44),  we  shall  have 


y  =  x'-ij 


(47) 


where 


d"n     ,     M,' d"-'Tj         M^  d^'-^Tj 


dx 


v  + 


dx 


;r^  + 


x''    dx"  -  ^ 


+ 


,  ^  ,      ii(n  —  i)  .  .  .{n  —  k-\-  i)    . 

Mk  —  — — -rr ^—^  r{r—\) 


M' 

?7  =  o  ; 

x""    ' 


ir-k^l) 


(48)     ^ 


+ 


(;/— 1)(«  — 2)  .  .  .  {ri  —  k-^\) 


r{r-\)  .  .  .  {r-k-^2)Mix) 


M„'=r{r—i)...   {r—7i-\-\)-{-r{r—  i)  .  .  .  {r—ti-\-2)AIlx) 


Now  if  (44)  admits  as  solution  the  value  of  y  in  (45),  then  (47)  must 
admit  a  solution  of  the  form 


(49) 


7  =  C  +  <^,-^  +  C.x''  + 


where  C^  is  necessarily  different   from  zero.     Before   going  farther 
take,  for  example,  the  case  n  =  2,  viz.: 


(50) 


d'y     ,    Al^dy    ,    M, 
dx'  ^  X   dx^  x'-^ 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE   REGULAR.     12$ 

Substitute  here  y  =  x''ij,  and  we  have 

d'^t]         2r{r  —  i)  4-  MJ^x)  drj 
^5^^     ~dx^  "^  ~^  ~  dx: 

r{r  -  I)  +  rMjx)  -{-Mix) 
+ -. V  =  O.. 

Now  this  must  have  an  integral  of  the  form 

(52)  V=  Q  +  C,x  +  6-,^^  +  .  .  .   ; 

where  C^  cannot  vanish.  This  value  of  t],  substituted  in  (51),  must 
give  a  result  which  is  identically  zero;  i.e.,  the  coefficient  of  every 
power  of  X  must  vanish.  Bearing  in  mind  now  that  M^  and  M^  are 
uniform,  and,  in  the  region  of  ;f  =  o,  continuous  functions,  it  is  clear 
that  X'  ^  is  the  highest  negative  power  of  x  that  will  appear  after 
making  the  substitution ;  the  coefificient  of  x~  ^  is  obviously  the 
quantity 

that  this  may  vanish  we  must  have,  since  C^  is  not  =  o, 

(53)  r{r  -  i)  +  rMSo)  +  Mid)  =  o. 

In  the  same  way  we  see  that  if  (49)  is  a  solution  of  (47)  we  must 
have  that  the  coefficient  of  x~  "  vanishes  ;  that  is. 


M„'(o)  =  o, 


or 


(54)         r{r  —  i)  .  .  .  {r  —  n -\-  i) -{- r{r  —  i)  .  .  .  (r  —  «  +  2)J/,(o) 
4_.  .  ,j^r{r-i)  .  .  .  {r-n  +  3)Mlo) 
+  .  .  .J^rM„_,{o)-i-M„{o)  =  o. 

This  is  the  same  as  equation  (25),  and  will  be  called  the  indicial 
equation.  If  r,-  is  a  root  of  the  indicial  equation,  and  i",-  the  corre- 
sponding root  of  the  characteristic  equation,  we  have 

27rt 
a  relation  which  has  already  been  shown. 


k 


126  LINEAR   DIFFERENTIAL  EQUATIONS. 

Notice  that  if  MJ^x)  =  o,  and  of  course  then  M„{o)  =  o,  the 
indicial  equation  is  divisible  by  r.  If  after  division  by  r  we  change 
r  into  r-\-  I,  we  obtain  the  indicial  equation  corresponding  to  the 

differential  equation  of  order  «  —  i,  which  is  formed  by  taking  -j- 

ax 

for  the  new  unknown  function. 

Suppose  a  system  of  fundamental  integrals  of  (44)  to  be  j, ,  jj/, , 

.  .  .  ,  j/„,  belonging  respectively  to  the    roots  r^ ,  r^,  .  .  .  ,  r„  o( 

the  indicial   equation.      We    will   further  suppose    that    the    roots 

r^  ,  .  .  .  ,  r„  are  so  arranged  that  none  of  the  differences 

r,  —  r,  —  I,     r,-  r,  —  I,     .  .  .  ,     r„  -  r,  —  i 

is  zero  or  a  positive  integer  :  this  arrangement  will  be  made  if  the 
real  part  of  none  of  the  roots  written  in  the  order  r, ,  r, ,  .  .  ,  r„  is 
greater  than  the  real  part  of  the  preceding  one. 

We  have  already  seen  that  if  a  regular  function,  say  F,  belongs 
to  an  exponent  r,  its  derivative  belongs  to  the  exponent  r  —  i;  also, 
if  two  regular   functions,  F  and   G,  belong  to  exponents    r  and  p 

F 

respectively,  their  quotient  -^  belongs  to  the  exponent  r  —  p. 

We  can  write  the  integrals  of  equation  (44)  in  the  form 

where  2  satisfies  a  linear  differential  equation  of  order  «  —  i.  If 
now_j/, ,  jj/, ,  .  .  .  ,  jj/„  belong  to  exponents  r, ,  r, ,  .  .  .  ,  r„ ,  and  if 
in  (44)  we  make  the  substitution 

J  =  yjzdx, 

then  the  integrals  (properly  chosen)  of  the  equation  of  order  «  —  i  in 
z  will  belong  to  the  exponents 

r,  —  r,  —  I,     ^3  —  r,  —  I,     .  .  .  ,     r„  —  r,  —  I  ; 

or,  in  other  words, 

r,  —  r,  —  I,     ^3  —  r.  -  I,     .  .  .  ,     r„  —  r.  —  i 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE  REGULAR.     12/ 

are  the  roots  of  the  indicial  equation  corresponding  to  the  differen- 
tial equation  in  z.     Again:  if  we  write 

the  roots  of  the  indicial  equation  corresponding  to  the  differential 
equation  of  the  order  n  —  2  in  t  will  be 

'3  '1  ^)        '4  '2  ^>        •    •    •    J        '»  'a  i> 

and  so  on.  Suppose  now  that  the  roots  of  the  indicial  equation  (54) 
or  (25)  are  arranged  in  the  prescribed  order,  viz.,  such  that  the 
differences 

(55)  ^  -  ^  —  I,     ^3  —  ^,-1,     •  .  •  ,     r„  —  r,  —  I 

are  neither  zero  nor  positive  integers  ;  then,  The  differential  equation 
(44)  admits  in  the  region  ofx=-oa  regular  integral  of  the  form 

y  =  x^^  0(.r)  ; 

* 
i.e.,  (p{x)  is  in  the  region  of  ;tr  =  o  a  uniform  and  continuous  function 
of  X,  and  does  not  vanish  for  ^  =  o. 

In  order  to  prove  this,  make  in  (44)  the  substitution 

(56)  y  =  x'-^rj, 
and  we  have 

d"7^        N,  d'^-'v         N,  d^^^-ri  N„ 

The  functions  N^  ,  N^ ,  .  .  .  ,  N„  _  ,  differ  only  from  ^//,  M^',  .  .  .  , 
M'„  _  I  of  equations  (46)  by  the  replacing  of  r  by  r, ,  and  by  making 

(58)  N.i.)  =  'J^. 

X 

If  we  write,  for  brevity, 


128  LINEAR  DIFFERENTIAL   EQUATIONS. 

equation  (57)  can  be  thrown  into  the  form 

(55)      ..  -  .^  +  Ni.).'  -i^+---  +  ^-.io)'£ 

=  N/(x)x"  -  '  , ^  +  N,\x)x''  -  ' ^- 

'^  '         dx''-'  ^      "  ^  ^  dx''-^ 

+  .  .  .+N'„_ix)x'^£+NXx)v- 

We  have  to  show  that  this  equation  admits  in  the  region  of  ;r  =  ©■ 
an  integral  of  the  form 

(60)  7  =  C  +  C,x  +  C,;r'  +  .   .   .  ; 

substituting  this  value  of  if  in  (56),  and  equating  the  coefficients  of 
x^  where  k  is  an  integer,  we  have 

(61)  [{k+i)k{k-i)  .  .  .  (^-;.+2)  +  A^Xo)(^'+i)^'(^-i)  .  .  .  (^'-^^+3) 
J^ Nlo){k-\-i)k{k  -  I)  .  .  .  (^^  -  «  +4)+  .  .  .  +  A/-„_,(o)(X'+i)]G  +  , 

where  the  coefficients  A„ ,  A^ ,  .  .  .  ,  Ak  are  made  up  from  mere 
numerical  quantities  and  from  the  coefficients  of  the  different 
powers  of  x  in  the  development  of  the  functions 

n:{x),  n:{x),  . . . ,  N'„_  ix),  N^{x). 

The  coefficient  of  CkJ^i  equated  to  zero  gives 

(62)  {k^i)k{k-i)  .  .  .  {k-n-{-2)-]-N,{o){k-\-i)k{k-i)  .  .  .  (k-n-^^) 

This  is  simply  the  indicial  equation  corresponding  to  equation  (57), 
divided  through  by  r,  and  then  r  changed  into  k-\-i.  Now  the 
roots  of  this  equation  have  been  shown  to  be  r,  —  r,  —  i,  .  .  .  , 
r„  —  rj  —  I,  none  of  which  are  either  zero  or  positive  integers;  there- 
fore, since  /^  is  a  positive  integer,  no  such  equation  as  (62)  can  exist, 
and  consequently  the  coefficient  of  C^  +  i  can  never  vanish  for  any 
positive  integer  value  of  k,  nor  for  ^  =  o. 

From  (61)  we  can  now  obviously  determine  each  coefficient  C  in 
the  form 

(63)  Ca    =    Ba<^„  . 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE   REGULAR.      129 

We  have  now  to  establish  the  convergence  of  this  series 

V  =  C^-\-  C,x  +  C^x^  +  .  .  . 

in  the  region  of  jc  =  o ;  for  this  purpose  we  compare  it  with  another 
aeries,  the  convergence  of  which  is  readily  ascertained. 

Let  jQ, ,  /^2>  •  •  •  >  ^n-xi  ^n  denote  the  maximum  moduli  of 
N^\x),  .  .  . ,  N' „  _  i(x),  N,lx) ;  then,  by  a  known  theorem  in  the  theory 
of  functions. 


(64) 


mod. 


mod. 


mod. 


£1. 


'd^{x)' 


'd''N„{xj 


<  a\ 


<  ^1  ^^, 
-I  p- 


<  a\ 


n. 


ix°-      J^  =  o      —  9° 


where  p  denotes  the  distance  from  the  origin  to  the  next  nearest 
critical  point.     We  form  now  the  auxiliary  differential  equation 


(65)  y.x' 


dx"  - ' 
d"  -  'V 


X 

I  —  - 
P 


X^' 


dx 


^  +  - 


y,x 

n 


•«  -  3 


d" 

i 

dx 


X'' 


I      — 


d"  -  ^z> 
dx"  -  ^ 


+ 


+ 


dv 

dx 

n„ 


I  — 


,       dv 
■r      dx 


+ 


P 


X 

I 

p 


v; 


where  the  quantities  y, ,  .  .  .  ,  y„  _  ^  are  arbitrary  positive  quanti- 
ties subject  merely  to  the  condition  that  no  root  of  the  equation 

(66)     y^  w{w  ~  \)  .  .  .{w  —  n-^^i)-^  y^wiw  —  i)  .  .  .  {w  —  n  +  4) 

+  ...  +  >/«- 1  =  o 

shall  be  either  zero  or  a  positive  integer.     We  seek  now  to  satisfy 
this  equation  by  a  series 


(67) 


V   =:    ^o-   x°- 


130  LINEAR  DIFFERENTIAL   EQUATIONS. 

where  g^  is  not  zero.  As  above,  substitute  this  value  in  (65),  and 
equate  coefficients  of  x^  :  we  have  without  difficulty 

(68)  \yik^  i)k...{k-n-^i)^  r#+  i)/^  .  .  .  (/^  -  «  +  4)  .  .  . 

where  the  coefficients  B  are  formed  in  the  same  way  as  the  coeffi- 
cients^ in  (61);  viz.,  the  ^'s  are  formed  out  of  the  coefficients  in  the 
developments  of 

x'  '  '  '  '  ^' 

I I 

p  p 

according  to  ascending  powers  of  x,  and  out  of  certain  numerical 
quantities,  just  as  the  A's  are  formed  from  the  coefficients  in  the 
developments  of 

N/{x),     N,\x),     .  .  .  ,     N„{x), 

and  the  same  numerical  quantities.  Since  the  ^'s  are  obviously  all 
positive  it  follows  from  (64)  that 

(69)  Ba  >  mod.  Aa,     a  =  I,  2,  .  .  .  ,  k. 

Equation  (68)  gives  now 

(70)  ga  =  Bag,  , 

where  iJSa  is  a  positive  quantity ;  if  then  we  take  g,  positive,  as  we 
obviously  may,  all  the  coefficients  g^,  g^  ,  •  •  •  will  be  positive.  It 
is  obvious  that  there  must  exist  a  finite  limit  for  k,  say  k  ^=  t,  such 

that  for  k  _  t  we  h-ave  always 

(71)  mod.  [/'(/&- 1)  .  .  .  {k-71-^^2)  +  N,{o)k{k-i)  .  .  .  (/^-;^+3) 

+  ...  +  iV„_,(o)] 
>  y,k{k-  I)  .  .  .  {k-n^2)  +  y^kik-  i)  .  .  .  {k-n+z)  +  •  •  •  r«  -  .  • 

Now  from  (61),  (68),  (69),  and  (71)  it  is  clear  that  we  shall  have 

gk>  mod.  Ck, 

if  this  can  be  shown  to  hold  for  k  /. 


ilquations  all  of  whose  integrals  are  regular.    131 
Let  A  denote  the  largest  modulus  in  the  series 
I,    Hi ,     H2 »    •  •  •  >   H/, 
and  let  B  denote  the  least  of  the  quantities 

I,     IB, ,      IBa ,     •  •  •  ,    !B/> 

and  suppose  we  choose  C^  so  that 

(72)  A  mod.  C,  <  B^„ . 

Since  A,  B,  ^^  are  different  from  zero,  this  inequality  is  of  course 
always  possible,  and  gives  for  C^  a  value  different  from  zero.  It  is 
clear  now  from  (63),  (70),  and  (72),  remembering  the  hypothesis  as 

to  A  and  B,  that  for  k  _  t,  and  consequently  for  all  values  k,  that 

we  have  always 

gk  >   mod.  Ck  . 
If  then  the  series 

(73)  ^    =  .^0    +  .^:^   +  ^.^"    +    .     •     . 

is  convergent,  it  follows  a  fortiori  that 

(74)  7  =  ^0  +  C,x  +  C,x'  +  .  .  . 

is  convergent.     To  establish  the  convergence  of  (73)  in  the  region  of 

X 

X  =  o,  we  proceed  as  follows:  Multiply  out  (65)  by  i ,  and  then 

P 
substitute  the  above  value  of  v\  this  gives,  on  equating  the  coeffi- 
cients of  x^, 

(75)  \{k^i)k{k-i)  ...{k-n-\-  3)r,  +  •  •  .  +  (^+i)r«-.]<r*  +  . 

=  _k{k-  i)...{k-n^2){r.>^^a) 


132  LINEAR   DIFFERENTIAL   EQUATIONS. 

We  have  from  this,  for  the  Hmit  of  the  ratio  of  two  consecutive  coef- 
ficients ^i+,,  gk, 

k  =  ^     gk  PYx 

and  for  the  Hmit  of  the  ratio  of  the  two  corresponding  terms  in  the 
series  (73) 

If 

mod.Zl±^;r,     =Zl±^mod.;r,  <i, 

py.  pyi 

then  the  series  is  convergent ;  that  this  may  be,  we  need  only  h'mit 
X  by  the  inequality 

py. 


mod.  X  < 


r.  +  P^i 


Since  y^  and  p  are  positive  non-vanishing  quantities  it  is  obvious 
that  (73)  has  a  circle  of  convergence  having  ;r  =  o  as  its  centre.  It 
follows  at  once  that  (74)  is  convergent  inside  this  same  circle.  As  ;/^ 
can  be  taken  as  large  as  we  please,  we  have 

Hm  -    ^^'        =  p, 

so  that  it  is  clear  that  the  series  (74)  is  convergent  in  the  entire 
region  of  ;ir  =  o.  This  can  also  be  shown  by  a  well-known  process  in 
the  theory  of  functions.  We  have  shown  now  that  (59)  admits  as  a 
solution  the  uniform  and  continuous  function 

t^  =  C,  ^  C,x  -{-  C,x'  +  .  .  .  , 

and  it  therefore  follows  that  (44)  admits  the  solution 

jj/^  =  x'-n^,         or,  say,         j,  =-  x'-^cpix)  ; 

where  (p{x)  is  a  uniform  and  continuous  function  of  x  in  the  region 
of  X  =  o.  We  have  now  shown  that  the  given  equation,  (44),  pos- 
sesses one  regular  integral;  the  remainder  of  the  theorem,  viz.,  that 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE  REGULAR.     1 33 

■all  of  the  integrals  of  (44)  are  regular,  is  quickly  proved.  If  in  (44) 
we  write 

y  =  yj^dx, 

we  find,  as  already  shown,  for  z  an  equation  of  order  n  —  I,  which 
in  form  is  the  same  as  (44).  The  roots  of  the  indicial  equation 
■corresponding  to  the  z  differential  equation  are 

r,  —  r,  —  I,     ^3  — r,  —  I,     ...    ,     r„  —  r,  —  I  ; 

denote  these  for  a  moment  by 

Pi  1        Pi  '        •     •     •    >        Pn  1 

then,  since  r, ,  r, ,  .  .  .  ,  r^  are  so  arranged  that  for  ft  >  a  the  differ- 
ence r^  —  Ta  is  never  a  positive  integer,  it  follows  that  the  same 
property  exists  for  the  roots  p^,  p^,  .  .  .  ,  p„;  viz.,  p^  —  p^  is  for 
ft  >  a  never  a  positive  integer.  The  differential  equation  in  z  there- 
fore admits  a  regular  integral  of  the  form 

^i  =  -^**  -  ^1  -  ^ip{x), 

ip{x)  being  a  uniform  and  continuous  function  of  x  in  the  region  of 
jtr  =  o.     Corresponding  to  z^  we  have 

y.  =  yj^.dx, 
an  integral  of  (44)  belonging  to  the  exponent 

^  -  ^,  -  I  +  I  +  ^1 ,     =  ^• 
By  the  same  process  we  can  find  a  third  regular  integral  of  (44), 

y.  =  yj^^dxf  t^dx, 

where  /,  is  a  regular  integral  of  a  differential  equation  of  order  n  —  2, 
and  belongs  to  the  exponent  r^  —  r^  —  i.  The  integral  y^  therefore 
belongs  to  the  exponent 

^3  -  r,  —  I  +  I  +  r,  -  r.  -  I  +  I  +  r, .     =  r, . 


134  LINEAR  DIFFERENTIAL  EQUATIONS. 

The  integrals  z^,  t^,  .  .  .  ,  w,  of  the  auxihary  differential  equations, 
are  always  of  the  form 

where  6{x)  is  a  uniform  and  continuous  function  of  x.  The  loga- 
rithms which  appear  in  the  integrals  of  (44)  can  then  only  enter- 
through  the  different  integrations  which  have  to  be  performed. 

We  have  thus  established  the  existence  of  a  fundamental  system 
of  regular  integrals  of  equation  (44),  the  elements  of  which  system 
belong  respectively  to  the  exponents  r, ,  r^ ,  .  .  .  ,  r„,  which  are  the 
roots  of  the  indicial  equation.  It  is  also  obvious  that  these  expo- 
nents are  respectively  equal  to 

-»■'         ^'j        •••>  „•» 

27Tt  2711  2711 

where  J", ,  ^^ ,  .  .  .  ,  s^  are  the  roots  of  the  characteristic  equation — 
a  result  arrived  at  in  what  precedes. 

Resume  for  a  moment  equation  (41),  viz., 

d^y  P,d"-'y        .  P„ 

— =—  -4-  — -  ■ -h  .  .  .A ^  =  o : 

dx"    ^    tp  dx"-^  ^  '    //'« -^ 

where  ip  =  x{x  —  x^{x  —  x^  .  .  .  {x  —  Xp),  and  consider  the  critical 

point  X  =^  X;.    We  can  show  that  the  sum  of  the  roots  of  the  indicial 

1        ....        .           ,          M?i  —  i)     ^  , 

equations  for  the  critical  points  is  equal  to  p— -.     Denote  by 

f\x,)  the  derivative  (— ^)  ;  the  indicial  equation  relative  to  this 

\dxl  X  —  Xi 

point  is  then 

{jG)     r{r-~  I)  .  .  .  (r  _  «+  I)  +  ^^  r{r  -  i)  .  .   .  {r  -  n  +  2) 

The  sum  of  the  roots  of  this  equation  is 

n{n  —  i)        fJi-^i) 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     I35 

Again :  the  indicial    equation    corresponding  to  the    critical   point 
jr  =  00  is 

r{r-\)  .  .  .  (r-;^+i)  +  K„.,-Or(r-i)  .  .  .  {r-n-{-2) -\- .  .  .  =  o. 
The  sum  of  the  roots  of  this  is 

n(n  —  i)  .      ,  n{n  —  i)         , 

2  2 

since  a,,^  „  _ ,  =  n  {ri  —  i). 

The  sum  total  of  the  roots  of  all  the  indicial  equations  is 

«(^Zj-   i)     ,      ,  V  Pj^h 

Now  by  a  well-known   formula  for  the  decomposition  of  rational 
fractions  we  have 

tp'{x^  X  —  Xi         ^{x) 
multiplying  this  by  x  and  then  making  ;r  =  cc,  we  have 

?(^~  ■' 

therefore,  finally,  we  have  for  the  sum  of  all  the  roots  the  value 

n{7i  —  i) 


In  concluding  this  chapter  we  will  give  a  few  illustrations  of  the 
development  of  the  integrals  in  series,  and  the  method  of  obtaining 
the  coefficients.  We  shall  use  for  convenience  only  the  point  ;ir  =  o, 
and  begin  with  an  equation  of  the  third  order,  viz., 

(J)  ^  ,  Qi^) ^ 

dx^        x{x  —  a^{x  —  a^  .  .  .  {x  —  a^)  dx"" 


+ 


x''\{x  —  a^  .  .  .{x  —  ay)Y  dx 


^  x'\{x  -a,)...{x-  a,)\'  ■" 


136  LINEAR  DIFFERENTIAL  EQUATIONS. 

where  the  critical  points  of  the  coefficients  are 

X  ■=  o,     X  =  a^ ,     .  .  • ,     X  =  a^ 

in  number  /^-j-  i  ;  and  Q.i-r),  Qy{x),  .  .  .  are  polynomials  in  x  of 
degrees  /i,  2yM,  .  .  .  respectively. 

Equation  (i)  may  be  put  into  the  following  form  : 

(2\  ^>  _,_  Pkx)   d^y        Pjx)  dy         Pjx) 

^  '  dx'~^      X      'dx'  ^     x""     dx~^     x'    -^ 

P^{x),  PJyX),  PJ^x)  being  rational  functions  of  x  for  which  ;ir  =  o  is 
not  a  pole. 

The  indicial  equation  is 

r{r  -  i){r  -  2)  +  Pio)r{r  -  i)  +  Ploy  +  ^3(0)  =  o, 

of  which  the  roots  {a,  b,  c)  will  first  be  supposed  all  different,  and  the 
difference  between  no  pair  of  them  an  integer.  Equation  (2)  will 
then  have,  in  the  region  of  the  point  x  =^  o,  three  integrals  of  the 
form 

x^'SciX^,     x^2Ci'x\      x''2cl' x\     {i  =  O,  I,  .  .  .  00.) 
Equation  (2)  may  be  conveniently  written 

Substituting  in  this  equation  the  series 

and  equating  to  zero  the  coefficient  of  x^^"^,  we  find 

(5)      Cy,F{}x^d)  +  f^  _  ,0,(/^+^—  0  +  ^/^  -  20,(^^+^—2)  +  .  .  .  =  O. 

The  symbols/^ and  0,,  0^ ,  .  .  .  have  here  the  following  meanings: 
The  point  x  =^  o  being  an  ordinary  point  for  P, ,  P^,  /*, ,  we  may 
write 

^,  =  «o  +  a^x  +  a^x'  +  .   .  .   , 

P,  =  ^„  +  b,x  +  b^x-^  +  .  .  .  , 

P,  =  d,-\-  d,x  J^d,x'+  .  .  .   ; 


1 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     1 37 
if  now  in  (4)  we  substitute  j/  =  x^ ,  then 

Fir)  =  r{r  -  i)(r  -  2)  +  ajir  -  i)  +  b,r  +  d, , 
0i(^)  =  a,r{r  -  i)  +  ^^r  +  <  , 
0,(r)  =  a,r{r  —  i)  +b,r^d,, 


')^,(r)  =  ayr{r  —  i)  -|-  ^^r  -|-  d,' . 


In  equation  (5),  making  //  =  o,  the  quantities  (:^  _,,...  all  vanish 
identically,  leaving  only  c^F{a)  =  o  ;  but  F{a)  =  o,  since  a  is  a  root 
of  F(r)  =  o ;  hence  r„  is  indeterminate. 
Making  /x  =  i,  we  have 

If  /i  =  2, 

(7)  ^.^(«  +  2)  +  c^(p,{a  +  I)  +  ^o0,(«)  =  O. 

li  M  =  3, 

^.H^  +  3)  +  ^.0/^  +  2)  +  c,(P,{a  +  I)  +  ^o03(^)  =  o , 


Hence 


^1  — 


Thus  every  c  of  index  >  o  can,  as  seen  above,  be  expressed  as  a 
product  of  c^ ,  which  cannot  =  o,  into  a  function  of  the  root  a,  and  so 

z'  =  CO 

the  series  y  =  2  c^x'^'^''  is  completely  known  when  any  value  arbi- 

z  =  o 

trarily  chosen  has  been  assigned  to  c„ .  In  a  manner  precisely 
similar  the  series  corresponding  to  the  roots  d  and  c  may  be  calcu- 
lated. 

Suppose  one   of  the   roots,  as  a,  to  be  zero  :  the  corresponding 


138  LINEAR  DIFFERENTIAL   EQUATIONS. 

i  =  CO 

integral  will  he  f  =  2  CfX',  which  on  being  substituted  in  equation 

I  =  o 

(4)  gives  for  the  coefficient  of  xf^ 

^^^{m)  +  ^m  -  i0i(A'  —  i)  +  ^M  -  20=0"  -  2)  4-  .  .  .  =:  o. 

Making  j^  =  o,  we  find 

c,F{o)  =  o,       but       F{o)  =  o ; 

hence  r,  is  arbitrary,  since  it  cannot  vanish.     It  may  be  noted  that 
P3  must  be  divisible  by  x  in  this  case,  or  d^  =  o. 
For  /<  =  I, 

^1^(0  +  ^-00X0)  =  o- 
For  M  =  2, 

^»^(2)  +  ^,0,(1)  +  ^O0.(O)  =  O. 

Therefore  the  coefficients  c  are  to  be  obtained  just  as  before.  The 
case  of  all  three  roots  of  the  indicial  equation  distinct  is  thus  com- 
paratively simple ;  but  a  numerical  example  may  not  be  found 
entirely  useless  in  this  connection. 

Suppose  the  coefficients  P^  ,  P^,  P^,  when  developed  by  Taylor's 
series  in  ascending  powers  of  x,  to  have  the  following  values  : 

P.=  i  +  ^^'  +  ^^'  +  ^'^''  +  .    •    •    , 

P.=  -i  +  d,x  +  cUx''  +  d^x^  +  .   .   .   . 

Thus  rt„  =  -^j  ^„  =  -^,  rt'^  =  —  \,  and  the  indicial  equation  for  ;ir  =  o 
becomes 

F{f)  =  r{r  —  i)(r  -  2)  +  -|-r(r  —  i)  +  ^r  —  -^  =  o,  or 
(6)  r^__VV+r-i  =  o. 

The  roots  of  this  cubic  are  i,  \,  and  \.  Since  no  two  of  them  differ 
by  an  integer,  there  are  no  logarithms  in  the  integrals,  and  we  may 


1 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     139 

obtain  them  by  putting  i,  i,  ^  successively  iox am.  equation  (5).  We 
have  then,  first, 

(7)  c,F{2)  +  r„0XO  =  o. 

But 

equation  (7)  therefore  becomes 

2i.,  +  (^,  +  <).„  =  o,     or     c,=  --^-^-c,. 

Again, 

Fia  +  2)  =  FiS)  =  3=  -  Y  X  3^  +  3  -  i  =  I3i,. 
0x(«  +  0  =  0.(2)  =  2^,  +  2^,  +  d, , 

0.(0  ^'^^  +  <; 

whence  equation  (7)  becomes 

I3i.,  -  {2a,  +  2^,  +  ^,)^tA)  ,^  +  ^^(^^  +  0  =  0; 
that  is, 

^.  =  f^|2(^,  +  <)(2^.  +  2^,  +  <)  -  5(/;,  +  01  . 

In  the  same  way,  as  many  of  the  coefificients  may  be  calculated  as 
are  desirable.  It  will  often  happen  that  c^ ,  which  is  a  common 
factor  to  all  the  terms,  may  be  so  chosen  as  to  simplify  the  series 
more  or  less. 

The  roots  ^  and  ^  may  now  be  treated  in  the  same  way,  and  each 
will  give  rise  to  a  convergent  infinite  series  all  of  whose  coefficients 
except  the  first  are  determinate. 

Returning  to  the  supposition  that  a,  b,  c  are  the  roots  of  F{f)  =  o, 
let  two  of  them,  as  a  and  b,  be  equal,  and  corresponding  to  the 
double  root  a  we  shall  have  two  integrals  ;  the  first  being  the 
convergent  infinite  series 

y,  =  2biX''  +  \     (?■=  I,  2,  .  .  .  ,  00,) 


I40  LINEAR   DIFFERENTIAL   EQUATIONS. 

and  the  second  of  the  form 

y^-=i  x^\'2  CiX'  -j-  2  c-x^  .  log  X } . 

Substituting  jj/^  in  equation  (4),  the  following  results  are  obtained 
for  the  coefficients  of  x^^^^  and  ^+'*  log  x,  which  must  separately 
vanish. 

Coefificient  of  x"'^'^ : 

(8)        F {a-\-/A)  c;  -\-(p,  {a-{-/x—  i)c ^  _  ,-j-(P^  {a-^pi—2y  ^  _  ^-\-  .  .  . 

Here  F'{a  +  ju)  denotes  the  result  obtained  by  differentiating  F{r)     j 

with  respect  to  r,  and  substituting  a  -\-  ^i  for  r  in  the  result ;  and     5 

similarly   for   the   other   functions,    0/,   0/,  ...      It  may  also    be 

noted  that  if  (p{x,  r)x''  be  the  result  of  substituting  x""  for  y  in  any 

linear  differential  equation,  the  result  of  substituting   x''  log^  x  is 

d^  d^ 

~T~K  i^i-^'  ^)'^''^]-     For  x*"  log-^jf  =  -j-jx'' ,  and  it  is  easy  to  see  that  it 

makes  no  difference  whether  we  perform  the  differentiation  A  times    ■ 
upon  x''  and  substitute  the  result,  or  substitute  x''  and  differentiate 
the  result  A  times  with  respect  to  r.     In   fact,   the  result   of  sub- 
stituting x''  is 

,  d''x'-  d"  -  'X'- 

where  the  coefficients  A,  B,  ...  do  not  contain  r,  and  since 

~dr^  ~dx^  ^^^'  ~dx^d?^  ^^^^' 

Hence,  knowing   the  result  obtained    by  substituting  x"" ,  that  for 
jr''  log^  X  is  given  by  the  following  identity : 

d'^  dcf)(x,  r) 

^  [(P{x,  r)x'-]  =  <f){x,  r)x'-  log^jr''  +  A  — -^—^^  log^  "  '^ 

A(A-J  ^V(^,_r)  _  d_^x^) 

+        2 !  dr'       -^  ^°^        ^  +  •    •    •  +       ^^K       ^  • 

If  A  =  I,  as  in  the  example  now  under  consideration,  the  formula 
ends  with  the  second  term. 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.      14I 

The  coefficient  of  x'^'^^  log  x  is  therefore 
(9)  F{a  +  yu)  c^  +  cp^a  +  A'  —  i)  ^V  -  i  +  02  («  +  /^  —  2)  ^V  -  2  +  •  •  • 
Making  yu  =  o  in  equation  (8),  we  find 

but  since  F{a)  and  F'{a)  are  both  equal  to  o,  c^  and  <:/  are  both 
arbitrary. 

In  the  same  equation,  making  yu  =  i,  2,  3,  .   .  .  ,  we  find 

c,F{a  +  I)  +  c.cpid)  +  .//^'(^  +  0  +  ^o>/(«)  =  O ; 
c,F{a  +  2)  +  ^.0,(«  +  I)  +  ^o0.(^)  +  c:F\a  +  2) 

+  ^/0i'(^  +  0  +  ^o'0/(«)  =  o  ; 

From  equation  (9),  in  like  manner, 
c:F{a)  =  o  ; 

From  the  latter  set  of  equations  the  coefficients  c/,  c/,  .  .  .  are  all 
obtained  in  terms  of  ^/,  which  is  still  undetermined. 

We  know  by  what  has  been  previously  shown    that  the  series 

Z'  =    CO 

-2  Ci  jf"  + ' ,  the  coefficient  of  log  x,  can  dilTer  only  by  a  constant  factor 

z'  =  o 

from  jj'j  =  SdiX^^'  (i  ^  o,  i,  .  .  .  ,  00).  The  root  d  of  the  equation 
F\?')  —  o  corresponds  to  a  series  entirely  similar  tojVi,  which  may  be 
found  in  the  manner  already  amply  illustrated. 

Let  us  now  suppose  that  the  three  roots  of  the  indicial  equation 
are  all  equal,  say  the  common  value  is  a. 

In  this  case  the  three  integrals  in  the  region  of  the  point  x  =  o 
are  of  the  following  forms:  ■'%. 

f,  =  x^  2  c/'x\     (z  =  o,  I,  .  .   .  ,  00  ;) 

J,  =  x''\2c/"x'  +  2c/'x'  .  log  A' I,     (z  =  o,  I,  .  .  .  ,  00  ;) 

j',  =  x''\:2 CiX'  4-  2 c/x' .  log  ;r  +  ^ c/'x' .  log'  x],{i  =  o,  I,  .  . . ,  00  .) 


142  LINEAR  DIFFERENTIAL  EQUATIONS. 

The  manner  of  finding  the  integrals  _j/j  and  y^  will  now  present  no 
difficulty  ;  as  to  y^ ,  remembering  the  useful  formula  given  above  for 
substituting  x''  log^;ir  in  the  differential  equation,  which  is  mentioned 
by  Jordan  {Coiirs  d' Analyse,  iii.  p.  82),  we  find  for  the  coefficients 
of  x"^^*^  log''  X,  xf-^*^  log  X,  x^+i^  the  following  expressions,  all  equal 
to  zero  : 

(10)  c"^F{a  +//)  +  ^'V  - 1  (p,{a-\-  /i- 1)  +  ^'V0.  («  +  /^-2)+  .  . .  =  o; 

(i  i)  c'^F{a  +  /^)4-^V  -  X  0.(^^  +  ;/  -  i)  +  ^V  -  2  0.(«  -\-  M—  2)+  . .  . 

+  ^'V  -  a  0/  («  +  At  -  2)  +  .  .  .  ]  =  o  ; 

(12)  c^F{a  +  /<)  +  r^  _  .  0,(^  +  >"  -  i)  +  ^M  -  20o.(«  +  //  -  2)  +  .  .  . 

+  ^-V  -  .  0/  (^  +  yu  -  2)  +  .  .  .  +  c\F"ia  +  //) 

+  f'V-,0,"(^  +  /'-   I)  +^'V-.0."(«  +  /'-2)+  ...   =0. 

Making  jj.  =1  o,  we  obtain  the  following  equations : 

c:'F{a)  =  o, 

c:F{a)  +  2c:'F'{a)=o, 

c,F{a)  +  c:F'{a)-\-c:'F"{a)  =  0', 

but  as  a  is  three  times  a  root  of  F{r)  =  o,  F{a),  F'{a),  F"{a)  all 
vanish  ;  and  since  c^ ,  r/,  r„"  cannot  vanish,  they  may  have  any  values 
whatever ;  the  integral  y^  therefore  contains  three  arbitrary  con- 
stants, the  maximum  number  for  an  integral  of  an  equation  of  the 
third  order.  Equation  (10)  enables  us  to  determine  all  the  remain- 
ing constants  r/',  c/',  .  .  .  ;  by  aid  of  them  equation  (11)  gives  r/, 
€^ ,  .  .  .  ;  and  knowing  these,  equation  (12)  gives  the  constants  c^, 
c^,  .  .  . 

The  integral  y^  is  thus  found  incidentally  while  calculating  j, 
(a  result  readily  foreseen  from  the  general  theory).  It  is  proper  to 
remark  that  upon  substituting  either  jj',  or  y^  in  the  differential  equa- 
tion, we  obtain  precisely  the  same  set  of  equations  for  determining 
the  constants  c^",  r/',  ...  as  already  found  by  substituting  j, . 

From  the  preceding  principles  and  examples  the  reader  will  de- 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     I43 

duce  with  ease  the  following  Rule  for  finding  the  integrals  of  a 
.linear  differential  equation  of  the  in^^  order  ivhen  they  are  all  regular  : 
Take  for  the  origin  that  critical  point  in  the  region  of  which  the 
■integrals  are  to  be  determined.  Multiply  the  equation  by  ;r"' ;  the 
origin  will  then  no  longer  be  a  critical  point  for  the  products /;ir, 
p^x^ ,  .  .  .  ,  p„,x"' .  Develop  these  products  by  Taylor's  theorem,  or 
■otherwise,  in  the  following  forms : 

A^     =  ^u    +  ^1-^  +  ^2^''  +  •  •  . 
A^^     =d„    -{-  b,x    +  iKX^   +   .   .   . 

P,nX"'  =  w„  4-  m^x  +  m^x''  +  .  .  . 

Form  the  indicial  equation, 

F{f)  r=  r{r  —  i)  .  .  .  {r  —  ti  -\-  i)  -\-  aj{r  —  i)  .  .  .  (r  —  «  -(-  2) 

+  ...+;;/„  =  O, 

and  obtain  its  roots.  If  no  two  roots  are  equal,  or  differ  by  an  in- 
teger, the  equation  has  m  integrals  of  the  form 

'2c,x^"^' ,     (/  =  O,  I,  .   .  .  ,  00,) 

where  a  is  one  of  the  roots.  If,  however,  two  or  more  roots  are 
equal,  or  differ  by  integers,  some  -of  the  integrals  will  in  general  con- 
tain logarithms  which  occur  in  a  manner  already  explained.  Let 
jr, ,  jFj ,  .  .  .  ,  7a  be  the  integrals  corresponding  to  the  root  a  of  mul- 
tiplicity A.  Substitute  y^  in  the  equation,  and  equate  to  o  the  co- 
efficients of  every  term  in  x  and  every  term  in  log  x  separately. 
From  the  system  of  equations  thus  formed  the  constants  of  the  series 
may  be  successively  determined,  with  the  exception  of  two  which 
will  remain  arbitrary ;  and  the  series  multiplying  log  x  will  be  the 
integral  J/,.  In  the  same  way  the  constant  coefficient  in  the  integral 
y  may  be  determined,  and  the  series  multiplying  log^;ir  will  again 
be  the  integral  j/j. 

Proceeding  in  a  similar  manner  with  each  root  of  the  indicial 
equation,  the  integrals  corresponding  to  the  region  of  the  point  ;r  =  o 
may  all  be  found. 


144 


LINEAR   DIFFERENTIAL   EQUATIONS. 


As  a  further  illustration,  let  us  obtain  the  integrals  of  the  equa- 


tion 


iVy 


dx 


1  + 


^  -\-  ^x"  -\-  x^ 


d'y 


+  ; 


?>x{x  -f-  0(-^  ~  ^X-^  ~  2)  dx' 
64  +  iGx'  +  4.x'  +  x'        dy 


+ 


Z2X'{X  +    lY{x  -    lY{x    -    2f  dx' 

512  +  64X'  +  8jr'  +  ;ir''        dy 
64x^\{x  -\-  i){x  —  i){x  —  2)Y  dx 

51200  -|-  3200;ir'  +  25^'" 
2048;tr' I  (;ir  +  l)(^  —   l)(^  —  2) 


J  =  O 


in  the  region  of  the  point  ^  =  0.     This  equation  is  seen  upon  exami- 
nation to  satisfy  the  conditions  that  all    of    its  integrals    shall  be 
regular,  and  each  numerator  is  of  the  maximum  degree  in  x. 
We  have 


/:(^) 


4X  -{-  2x''  -\-  x" 


81(^+  1){X  -  i){x  -  2)1 

=  i{x  +  x^-^lx^+  V-^^  +  3^^^  +  i^x'  4-  -w-^'  4- 


A^  = 


64  +  i6x'  -\-  4x*  +  x' 


32{{X  -i-    I){X   -    l){x-2yr 


/s-^'    =    - 


Si2-^64x'  +  ajr^+^t-" 


64j(;r+i)(^-  i)(;t--2)r 

/^4r*  _        5  J  200 -f  3200;tr^  +  25^' 

2048j(;ir  +  l){x  —  l){x  —  2)\* 

Hence 

a,  =  0,     b,  =  1      c,=  -I,     rt'^  =  ff. 

The  indicial  equation  corresponding  to  ;tr  =  o  for  an  equation  of 
the  fourth  order  is 


r(r-i)(r-2)(r-3)  +  a,r{r-\){r-2)  +  Kr{r-\)  +  c,r  +  d,  =  o, 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     145 
which  upon  reduction  becomes 

r'  +  K  -  ^y  +  {K  -3^0+1 0^'  +  (^0-  K  +  2^0-  6)r  +  ^0  =  o. 

Substituting  in  this  the  above  values  for  a^,  b^,  c„ ,  d^,  it  becomes 

r'  _  6^="  +  I  i^r'  -  7^r  +25^0, 

of  which  the  roots  are  found  to  be  ^,  J,  f ,  |,  the  second  pair  differ- 
ing from  the  first  by  the  integer  2.  We  can  now  obtain  an  integral 
of  the  following  form  : 

+  2rx^  +  ''{c,  +  C,  log  X  +  c",  log^  X  +  ^-V  log^  ^), 

For,  substituting  this  expression  in  the  differential  equation,  and 
placing  equal  to  zero  the  coefificients  of 

^  +  M  log'  X,        x^  +  f^  log'  X,        ;f  i  +  f'  log  X,        xi^f", 

we  obtain  the  following  equations : 

^a+/^)^"v+0,a+/^- lyv  -  .+0.(i+yu-2).''v -.+...  =  o;  (A) 

^a+y"KV  +0.(1+/'-  ly'v  - .  +0.a+yu-2y",  _.+... 
+3[^'a+/'W+0/(i+/^-  ly^v  -  .+0/(i+/^-2K'v  - . 

+  -..]=o;    (B) 

^(i+/^)^V  +  <^.{i+M-  iK  - 1  +  (pM+M-^y^  -.+  ... 

f  2[/^'(i+/.y  v+0/(i+y"-  0^'v  - .+  0;a+y"-2).^v  -.  +  •••] 
+3[^'Xi+/^K'v  +  0/'a+/^-iK'v  - .  +  0;'a+/^-2y".  - . 

+  ...]  =  o;(C) 
^(i  +  /^)^^  +  0,(i  +  >"  -  0^^  -  .  +  0.(i  +  yu  -  2)r^  _  ,  +  .  .  . 

+/^'(i+/.yv+0/(i+/^-i)^v  -  .+0;a+/^-2).v  -.+  ... 
+i^-a+/'>'v+0/'(*+y"-  ly  V  -  ,+0;xi+/^-2)^'v  -.  +  ••. 
+/^-a+/'W+0/"(i+/^-  ly  V  -  .+0/''a+/.-2).-v  - . 

+  .  .  .  =  o.  (D) 
The  meaning  of  the  symbols  F,  F' ,  .  .  .,  0j,  0/,  .  .  .  has  already 


14^  LINEAR  DIFFERENTIAL  EQUATIONS. 

been  explained ;  it  is  now,  however,  convenient  to  write  them  down 
in  full.     We  have,  viz.. 

Fir)  =  r'  -  6r'  +  ^r'  -  i/r  +  ff , 

F'{r)  =  4r'  -  i8r^  +  23/-  -  J/-, 

F"{r)  =  i2r'  -  36r  +  23, 

i^^'^r)  =  24r  -  36  ; 

<Pr{r)  =  Hr  -  ^){r  -  2)  +  ^r{r  -  i)-|r  +  ^ 

=  i^^  -  i^'  -  1^  +  ¥, 
^/(^)  =  1^'  -  i^  -  I, 
<p:'{r)  =  fr  -  i 

In  fact,  as  before  mentioned, 

(pm{r)  =  a,„r{r  —  i)(r  —  2)  +  b,„r{r  —  i)  +  c„,r  +  d„, 
=  a,y  +  {b,„  —  ^a„y  +  {c,„  —  b,„  +  2a„')r  +  ^,„ , 

0'«X^)  =  3«»^'  +  2('^,«  —  3^».)^  +  (^'«  +  ^»  —  2«,„), 
0,/'(^)  =  6rt,„r  +  2{b,^  —  3««), 
0„/-(r)  =  6a„r, 
a„,  b,„,  c,„,  d,n   being  the  coefficients  of  x'"-  in  the  expressions  of 
p^x,    p,x\    p,x\    p,x\ 

Directing  our  attention  to  equation  (A)  and  making  /<  =  o,  we  ob- 
tain only  the  identity  F{^)cJ"  =  o;  making  /x  =  i,  we  find 

Now  0,a)  =  sV  -  iV  -  I  +  ¥  =  H ;     F{ii)  =  I  If.     Therefore 

which  will  not  alone  suffice  to  determine  c^"  and  c^" .     However, 
making  /i  =  o  in  equation  (C),  the  result  is  zF"{^c^"  —  o.     Since 


EQUATIONS  ALL    OF    WHOSE   INTEGRALS  ARE  REGULAR.     I47 

t|  is  a  root  of  multiplicity  2  only,  F"{^  is  not  =  O  ;  hence  c^"  =  o, 
and  consequently  c/"  =  o.  By  making  >u  =  2  in  equation  (A),  we 
find  F{2i)  c^"  =  o,  whence  it  may  be  concluded  that  c^'"  is  arbitrary. 
In  equation  (B),  making  //  =  i,  we  obtain  i  i^c/'  -f-  H^o"  =  o,  and 
•making  yu  =  o  in  equation  (D),  F"{^)c^"  =  o  ;  hence  c/'  =  o  and 
^/'  =  o.  Assigning  the  value  2  to  /^  in  equation  (B),  the  result  is, 
since  F'{2^)  =  o,  F{2^)c^"  =  0,  showing  that  <r/' is  also  arbitrary. 
The  only  conditions  by  which  to  determine  c^  and  cj  are  the  follow- 
ing, derived  from  (C)  and  (B) : 

F{i)c:  =  o        and         F{^y,  +  F^iy/  =  o, 

both  of  which  are  identities,  so  that  c^  and  c/  are  also  arbitrary. 
Returning  to  equation  (A),  let  yw  =  3.     Then 

^i3iyr+<PMyr  =  o 

whence 

,„  _    _    0,(2i)       ,„ 

'     ~        ^(3i)     '    ' 

In  like  manner,  by  making  yu  =  4,  5,  .  .  .  ,  the  coefifilcients  c/'\  c^'\ 
.  .  .  may  all  be  found  from  equation  (A).  Aided  by  these  values, 
■c/\  c^' ,  .  .  .  are  to  be  obtained  in  like  manner  from  equation  (B), 
.<:/,  .  .  .  from  (C),  and  c^ ,  c^,  .  .  .  from  equation  (D).  Thus  we  have 
an  integral  of  the  form  announced  containing  four  arbitrary  con- 
stants ;  it  is  therefore  the  general  integral  of  the  given  equation  in 
the  region  of  the  point  ,r  =  o.* 

Among  functions  of  the  kind  considered  is  obviously  the  func- 
tion y,  defined  by  the  irreducible  algebraic  equation  /{:*:,  j)  =  o  of, 
say,  the  n^^  degree  in  jy.  If  the  n  branches  of  the  function  j/  {i.e., 
the  n  roots  of  /"  =  o)  so  defined  are  linearly  independent,  then  y 
satisfies  a  linear  differential  equation  of  the  above  form  of  the  n^^ 
order ;  if,  however,  there  are  only  m  {in  <  n)  linearly  independent 
branches,  it  is  clear  that  y  will  satisfy  an  equation  of  the  m^^  order. 
If  a  differential  equation  is  satisfied  by  a  particular  root  of  the 
irreducible  algebraic  equation /(;ir,  y)  =  o,  it  must  be  satisfied  by  all 
the  roots.     Since  the  remaining  roots  are  branches  of  the  one  func- 

*  The  preceding  illustrations  of  the  general  theory  are  due  to  Mr.  C.  H.  Chapman. 


148  LINEAR   DIFFERENTIAL   EQUATIONS. 

tion  y  obtained  by  travelling  round  certain  critical  points,  and  if  y^ 
were  the  chosen  integral,  it  will  remain  an  integral  during  all  the 
motion  of  the  variable,  though  the  branch  y^  changes  into,  say,  y^ , 
etc.  Suppose  the  given  differential  equation  to  be  of  order  n,  and 
suppose  that  among  the  n  branches  of  the  function  y  there  are  only 
ni  {in  <  li)  linearly  independent ;  then  the  functions  J, ,  .  •  •  ,  JF„ 
satisfy  a  differential  equation  of  order  ;;/,  and  so  this  equation  has 
only  algebraic  integrals,  and  the  given  equation  has  these  same  in- 
tegrals and  some  additional  ones.  The  first  equation,  having  all  the 
integrals  of  the  second  for  integrals,  is  said  to  be  a  reducible  equa- 
tion. We  have  here  the  first  notion  of  reducibility  and  irreducibility 
in  differential  equations,  the  notion  being  entirely  analogous  to  that 
of  reducibility  and  irreducibility  in  algebraic  equations.  This  sub- 
ject will  be  taken  up  later  on  for  a  fuller  discussion,  but  it  is  con- 
venient to  give  here  a  few  theorems  in  connection  with  the  notion 
of  reducible  equations.  Suppose  the  linear  differential  equation 
P  =:  o  has  among  its  integrals  all  of  the  integrals  of  g  =  o,  where 
(2  is  of  a  lower  order  than  P.  Since  among  the  integrals  oi  P  =  o 
there  are  functions  which  do  not  satisfy  ^  =  o,  it  is  clear  that  the 
equation  P  =  o  may  have  critical  points  which  do  not  belong  to 
Q  =  o.  Conversely,  in  spite  of  the  fact  that  the  integrals  of  ^  =  o 
are  all  integrals  of  P  =  o,  it  may  happen  that  Q  =  O  has  critical 
points  which  do  not  belong  to  P  =z  o.  It  is  easy  to  see  what  the 
character  of  the  indicial  equation  is  for  these  points,  and  conse- 
quently the  character  of  the  integrals  in  the  region  of  the  points. 
Suppose  a  one  of  the  critical  points  of  ^  ^  o  which  does  not  be- 
long to  P  =  O ;  now,  remembering  that  the  only  critical  points  the 
integrals  of  a  differential  equation  can  have  are  those  of  the  equa- 
tion itself,  it  follows  that  in  the  region  oi  x  ^=  a  the  integrals  of 
Q  =  o  must  be  uniform  and  continuous  functions,  since  otherwise, 
as  these  integrals  are  also  integrals  of  /*  =  O,  this  last  equation  must 
have  X  =  a  as  a.  critical  point,  which  is  contrary  to  hypothesis. 

In  the  region  of  a  non-critical  (or  neutral)  point,  say  a,  oi  Q  ^  o 
(or  of  any  other  linear  differential  equation),  the  m  integrals  will  be 
developed  in  series  of  powers  oi  x  —  a,  whose  first  terms  are  re- 
spectively 

{x-ay,     {x-aY,     {x-ay,     ...,     (;r -«)'«-■; 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     I49 

if  «  is  a  critical  point  of  ^  =  O  of  the  kind  just  mentioned,  the 
developments  of  the  integrals  in  positive  integral  powers  of  ;ir  —  a 
will  begin  respectively  with 

{x  —  df-^ ,     {x  —  df-^ ,     .  .  .  ,     {^  —  «)'"'" ; 

where  the  positive  integers  r^,  r^,  .  .  .  ,  r,„  are  all  different,  and  do 
not  coincide  with  the  numbers 

o,     I,     2,     .  .  .  ,     7n  —  I, 

and  can  consequently  not  all  be  less  than  in.  It  is  also  clear  that  if 
any  one  of  these  roots  r^,  r^,  .  .  .  ,  r,„  is  greater  than  n  —  i,  the 
point  X  =  a  is  a  critical  point  of  P  =  o,  and  that  the  integrals  in 
the  region  of  this  point  will  be  developed  in  positive  ascending 
powers  of  x  —  a,  the  first  terms  of  the  developments  being  as  above, 
{x  —  ay^ ,  (x  —  ay-i ,  .  .  .  Suppose  for  a  moment  we  call  these  points 
quasi-critical  points ;  we  have  then  for  m  <  n  the  theorem  :  If  a 
linear  differential  equation  of  order  n  has  among  its  integrals  all  the 
integrals  of  a  linear  differential  equation  of  order  in,  tJien  all  the 
critical  points  of  the  latter  equation  zvhich  are  not  critical  points  of 
the  first  are  quasi-critical  points,  and  the  roots  of  the  indicial  equations 
belonging  to  S2ich  points  are  all  integers  each  of  which  is  less  than  n, 
the  order  of  the  given  equation. 

Among  the  different  branches  of  an  integral  of  a  given  differen- 
tial equation  (uniform  coefficients  always  understood)  of  order  n 
there  can,  of  course,  exist  at  most  only  n  which  are  linearly  inde- 
pendent ;  if,  however,  there  are  only  in{in  <  «)  (it  will  be  assumed 
hereafter,  unless  something  is  said  to  the  contrary,  that  m  is  always 
<  n)  linearly  independent  branches  of  the  function,  it  will  satisfy 
an  equation  of  order  in,  and  consequently  the  given  equation  is 
reducible.  It  follows  conversely  from  this  that  the  number  of 
linearly  independent  branches  of  a  function  which  is  an  integral  of 
an  irreducible  linear  differential  equation  is  exactly  equal  to  the 
order  of  the  equation.  It  is  also  evident  that  among  the  integrals 
of  a  reducible  equation  there  are  always  some  the  number  of  whose 
linearly  independent  branches  is  less  than  the  order  of  the  equation. 

Again,  if  a  function  y  is  an  integral  of  a  given  equation,  then  all 
of  its  branches  are  integrals  of  the  same  equation.  Suppose  the 
given  equation  to  be  an  irreducible  one  of  order  in;  then  among  the 


I50  LINEAR  DIFFERENTIAL  EQUATIONS. 

branches  of  y  there  are  just  in  linearly  independent  ones,  viz., 
jj/,  ,j>/, ,  .  .  .  ,j,„.  If  now  jj/ satisfies  another  equation  of  the  same 
kind  but  of  order  n,  then  J, ,  J^ ,  •  .  .  ,  ym  satisfy  it,  and  conse- 
quently, denoting  by  c^,  c^,  .  .  .  ,  c,^  arbitrary  constants, 

y  =  ^1^1  +  ^»j.  +  •  •  •  +  <^«.y,. 

also  satisfies  it.  But  Y  is  the  general  integral  of  the  irreducible 
equation ;  and  so  it  follows  that  if  a  given  equation  has  for  an  in- 
tegral one  of  the  integrals  of  an  irreducible  equation,  it  has  among 
its  integrals  all  of  the  integrals  of  the  irreducible  equation.  Suppose 
now  that  we  have  a  reducible  linear  differential  equation  of  order  n; 
it  must  have  an  integral  y  in  common  with  an  equation  of  lower 
order  ;«:  and  suppose  that  among  the  branches  of  the  function  y 
there  are  /  which  are  linearly  independent ;  then  /  can  be  at  most 
equal  m.  From  what  has  been  said,  we  see  that  y  satisfies  a  differ- 
ential equation  of  order  /,  of  which  the  general  integral  is 

y  =  ^  J'l  +  ^.  A  +  •  •  •  +  07/  • 

Now  every  differential  equation  which  is  satisfied  by  y  must  be  satis- 
fied by y^,y^,  .  .  .  ,yi,  and  consequently  V  satisfies  the  reducible 
differential  equation  of  order  ?i.  If,  therefore,  a  given  linear  differ- 
ential equation  is  reducible,  there  exists  a  linear  differential  equation 
of  lower  order  all  of  whose  integrals  are  integrals  of  the  given  equa- 
tion. As  this  last  equation  may  be  again  reducible,  we  have  that  if 
a  given  linear  differential  equation  is  reducible  there  must,  as  is 
easily  seen,  be  one  or  more  irreducible  equations  all  of  whose  in- 
tegrals are  integrals  of  the  given  equation. 

We  have  seen  that  if  an  integral  of  a  linear  differential  equation 
changes  into  itself,  multiplied  by  a  constant  s  when  the  independent 
variable  turns  round  a  critical  point,  j'  is  a  root  of  a  certain  alge- 
braic equation,  viz.,  the  characteristic  equation  corresponding  to 
the  critical  point.  If  .f  is  a  simple  root  of  this  equation,  there  is 
only  one  integral  satisfying  the  condition 

Sy  =  sy; 

but  if  ^  is  a  multiple  root  of  the  characteristic  equation,  say  of  order 
J^  -f-  I,  then  there  are  corresponding  to  it  /^  -|-  i  independent  integrals. 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     151 

If  these  are  regular  integrals,  we  know  from  what  precedes  that 
their  general  form  is,  in  the  region  of  ;tr  —  a, 

J  =  00  +  01  log  {x  —  d)-\-  0,  log'  {x  —  a) -\-  .  .  . -\- (t)^  Iog«  {x  —  a)\ 

where  0„ ,  0, ,  .  .  .  are  functions  which  may  vanish  for  x  ^=  a,  and 
which  when  x  turns  round  the  point  ;r  =  «  change  into  themselves 
each  multiplied  by  s.  The  highest  exponent  of  log  {x  —  a)  which 
can  enter  into  this  group  of  integrals  cannot  be  greater  than  >^;  if  this 
highest  exponent  is  equal  to  k,  then  there  can  exist  but  a  single 
integral  corresponding  to  ^  which  contains  no  logarithm.  This  has 
already  been  seen,  but  it  is  convenient  here  to  prove  it  in  another 
way.* 

Suppose  ya,  to  be  an  integral  corresponding  to  the  root  s  of  the 
characteristic  equation  for  the  point  ;ir  =  «  ;  then  ja  is  of  the  form 

y„.  =  0a,  o  +  0«,  I  log  (-^  —  ^)  +  •  •  •  ^  ^0.^0.  log"  {x  —  a); 

where  0a,  a  is  of  course  not  zero.  If  we  apply  the  substitution  S, 
i.e.,  make  x  turn  round  the  point  ;i:  =  rt:,  the  result,  Sya.,  is  again  an 
integral  of  our  equation,  and  is  of  the  form 

yo.-^27tiay^_^, 
where 

Ja  -  I  =   0a  -  1,0+  0a  -  I,  I   log  (;ir  —  rt-)  +  .   .  .  +  0„  _  ,,„  _  I  log*  -  ^  {x  —  O), 

and  where 

0a-i,a-i   ^(Pa.0.; 

and  consequently  ^a-i,a-i  is  not  zero. 

Now  since  _ya  +  27tiay^_  ^  is  an  integral  of  the  given  equation, 
it  is  clear  that  ja  _  i  is  also  an  integral.  It  follows  now  that  if  to  the 
root  s  of  the  characteristic  equation  for  the  point  x  =^  a  there  is  an 
integral  in  which  the  highest  power  of  log  [x  —  a)  is  log"  [x  —  a), 
there  exist  integrals  in  which  (S  is  the  exponent  of  the  highest  power 
of  log  {x  —  a),  where  ^  is  any  one  of  the  numbers  o,  i,  2,  .  .  .  ,  ar  —  i. 

*What  immediately  precedes  and  follows  is  taken  directly  from  an  important 
memoir  by  Frobenius — Ueber  den  Begriff  der  Irreductibilitdt  in  der  Theorie  der  littearen 
Differentialgleichungen  (Crelle,  vol.  76).  This  memoir  will  again  be  referred  to 
later  on. 


152  LINEAR  DIFFERENTIAL  EQUATIONS. 

We  see  now  that  if,  corresponding  to  the  root  s  of  the  charac- 
teristic equation,  there  is  an  integral  in  which  k  is  the  exponent  of 
the  highest  power  of  log  {x  —  a),  there  exists  a  group  of  ^  -|-  ^ 
linearly  independent  integrals  having  as  exponents  of  the  highest 
power  of  this  logarithm  o,  \,  2,  .  .  .  ,  k\  say  these  integrals  are 
y^,  y^i  •  '  •  , yw  Now  since  to  a  /^  -|-  i-fold  root  of  the  characteristic 
equation  there  can  exist  only  k  -\-  i  linearly  independent  integrals, 
it  follows  that  if  x  turns  round  the  point  x  ^  a,  any  one  of  these 
integrals,  say  y^ ,  will  become 

Syo  =  ^0  Jo  +  ^.  J.  +  •  •  •  +  ^kyk ; 

equating  the  coefficients  of  the  different  powers  of  log  {x  —  a),  we 
have 

Ck  =  (^k-i=    •    .    •    =  c^=  o, 
and  consequently 

We  will  say  that  two  integrals  whose  ratio  is  a  constant  do  not 
differ  from  each  other;  e.g.,  y^  and  Sy^,  =  c^y^,  do  not  differ  from 
each  other. 

Suppose  now,  conversely,  that  the  given  equation  has  only  one 
integral  in  the  region  oi  x  ^=-  a  which,  when  the  substitution  6"  is 
applied,  changes  into  itself  multiplied  by  the  {k-\-  i)-fold  root  s  of  the 
corresponding  characteristic  equation :  we  must  have  then  that  k  is 
the  exponent  of  the  highest  power  of  log  [x  —  a)  which  can  enter  into 
any  of  the  integrals  belonging  to  the  root  s.  Suppose  j„  to  be  the 
integral  considered  ;  then  Sy^  =  sy^.  If  ^  >  o,  the  equation  has 
other  integrals  than  y^  which  correspond  to  the  root  s ;  and  since  y^ 
is  the  only  integral  satisfying  the  relation 

Sy,  =  sy, , 

these  other  integrals  must  involve  logarithms,  and  one  of  them,  say 
j/j ,  will  be  of  the  form 

y,  =  01,0+  0,,i  ^og{x  —  a). 

Now  we  know  that  0i_ ,  can  only  differ  from  y„  by  a  constant  factor, 
and  of  course  by  choosing  j/j  properly  this  factor  maybe  made  unity, 
and  so  we  have  (p^^  ^  =  jj/„ ,  and  consequently 

Ji  =  (pi.o-\-yo'^og{x  —  a). 


EQUATIONS  ALL    OF    WHOSE  INTEGRALS  ARE  REGULAR.     153 

Suppose  now  there  exists  a  second  integral  in  which  log  {x  —  a) 
enters  to  the  first  power,  say  j/ ;  it  must  have  the  form 

y!  =  0'i,o+7olog(^-«). 
We  have,  therefore, 

y!  -y^  =  0'i,o-  0.,o; 

but  the  difference,  jj//  —  j, ,  is  an  integral  of  the  equation,  and,  as  it 
contains  no  logarithms,  we  must  have 

y^  —y.  =  ^0  Jo ,     or    //  =  c,y,  -\-y, ; 

where  c^  is  a  constant.  It  follows  then  that  every  integral  which 
only  contains  the  first  power  of  log  {x  —  «)  is  a  linear  function  of  y^ 
and  j/j .  If  /^  >  I,  then  corresponding  to  the  {k -\-  i)-fold  root  s 
there  must  be  more  than  two  linearly  independent  integrals,  and 
among  them  there  must  be  one,  sayjj/^,  of  the  form 

y^  =  02, 0  +  02, 1  log  {x  —  a)-\-  0,, ,  log'  {x  —  a). 

It  can  be  shown  just  as  before  that  all  integrals  which  contain 
log(;tr —  a)  only  to  the  second  power  are  linearly  expressible  in  terms 
<^f  Jo  5  Ji )  and  jFj.  By  continuing  this  process  we  see  that  if  to  the 
root  s  of  order  of  multiplicity  k -\-  i  there  corresponds  only  one  in- 
tegral free  from  logarithms,  then  k  is  the  exponent  of  the  highest 
power  of  log  {x  —  a)  which  can  appear  in  the  group  of  integrals  be- 
longing to  the  root  s.  We  arrive  thus  at  the  theorem  :  In  order  that 
a  linear  differential  equation  shall  possess  only  a  finite  number  of  in- 
tegrals which  satisfy  the  co7idition  Sy  =:  sy  {where  S  refers  to  any 
critical  point,  and  s  is  any  root  of  the  corresponding  characteristic 
equation),  it  is  necessary  and  sufficient  that  the  roots  of  the  characteristic 
equation  corresponding  to  the  critical  point  in  question  shall  be  all  dis- 
tinct, or,  ill  the  case  of  a  root  of  order  of  multiplicity  k  -\-  \,  that  k 
shall  be  the  exponent  of  the  highest  power  of  the  logaritJun  in  the 
integrals  of  the  group  belonging  to  this  root. 


CHAPTER  VI. 

LINEAR   DIFFERENTIAL   EQUATIONS   OF   THE   SECOND   ORDER, 
PARTICULARLY   THOSE  WITH   THREE   CRITICAL  POINTS. 

The  linear  differential  equation  of  the  second  order  with  three 
critical  points  is  the  one  which  has  been  studied  more  than  any- 
other,  and  is  of  the  highest  importance,  as  its  integrals  include  many 
of  the  most  important  functions  with  which  mathematicians  are,  as 
yet,  thoroughly  familiar.  Before  taking  up  the  case  of  three  critical 
points  it  will  be  convenient  to  prove  a  theorem  due  to  Fuchs^  con- 
cerning a  transformation  of  differential  equations  of  the  second 
order — a  transformation  which  does  not  apply  to  equations  of  a 
higher  order.  As  shown  in  the  last  chapter,  the  most  general  form 
of  a  differential  equation  of  the  second  order  with  p  finite  critical 
points  and  all  of  whose  integrals  are  regular,  is 

where 

at)  =  (/-/,)(/-/,)  .  .  .  {t-i,), 

and  where  P^t),  PJ^t)  are  polynomials  of  the  degrees  p  —  i  and 
2/3—2  respectively.  Fuchs  proves  that  such  an  equation  can  by  a 
change  of  the  dependent  variable  z  be  thrown  into  the  form 

(2)  m^+QAt)'^^-Qlt)y  =  o', 

where  the  degrees  of  tp,  Q, ,  and  Q^  are  p,  p  —  i,  and  p  —  2  respec- 
tively. 

*Heffter:  Inaugural-dissertation.     Berlin,  1886. 

154 


EQUATIONS  OF  SECOND   ORDER— FUCHS' S  TRANSFORMATION.  I55 

Suppose  r,-  a  root  of  the  indicial  equation  corresponding  to  the 
critical  point  ti ;  also  suppose  that  r,-  is  not  zero.  The  indicial  equa. 
tion  is 

(3)  ^('' -'>  +  '•  ?(0  +  no  -  °- 

Make  the  substitution 

(4)  z  =  n{t-t>iy  ;     (/  =  I,  2,  .  .  .  ,  p;) 

equation  (i)  becomes  now 

+ 1  ^i^  Lf  -(J377f  +, , ,  ,^ ,  (Tr7:)r/  - 1,)\ 

Dividing  through    by  ^71  (/  —  /j)  we  see  at   once   that  the    coef- 

ficients  of  -~  and  ~  have  the  required  form.     That  the  coefficient 
df  dt  ^ 

of  y  has  the  required  form  can  be  seen  by  aid  of  (3),  or  directly  as 
follows:  If  we  divide  first  by  Hit  —  /,)  the  coefficient  of  y  is  seen  to. 
be  an  integral  function  of  /  of  the  degree  2p  —  2.  Denote  this  for 
the  moment  by  P^it).  In  consequence  of  the  transformation  (4) 
there  will  be  one  root  of  each  of  the  indicial  equations  correspond- 
ing to  /j ,  .  .  .  ,  /p  which  is  zero,  and  therefore 

^p^)=o;     (.=  1,2,. ..,p;) 

and  so  /*/(/)  is  divisible  by  tp{i),  and  the  quotient  is  a  polynomial  of 
degree  p  —  2.  This  transformation  cannot  be  made  when  all  the 
roots  r,-  are  zero  ;  but  in  this  case  the  differential  equation  has. 
already  the  required  form. 


t$6  LINEAR   DIFFERENTIAL   EQUATIONS. 

The  differential  equation  of  the  second  order  may  now  be  writ- 
ten in  the  simpler  form 

"where  /^,(/)  and  /*„(/)  are  polynomials  of  the  degrees  p  —  i  and  p  —  2 
respectively,  and  ^'(/)  —  {t  —  t^  .  .  .  {t  —  /p).  We  will  take  up  now 
the  case  of  only  three  critical  points  ;  it  will  be  convenient  to  obtain 
directly  the  result  of  the  above  theorem  for  this  particular  case 
{Jordan,  Cours  d' Analyse,  vol.  iii.  p.  220). 

Suppose  the  given  differential  equation,  say  /*  =  o  (where  z  is 
the  dependent  and  /  the  independent  variable)  has  three  critical 
points,  and  denote  by  /  the  independent  variable ;  then  the  critical 
points  may  be  denoted  by  /,  ,  t^,  t^.  We  will  first  transform  the 
equation  by  the  relation 

There  are  here,  of  course,  only  three  arbitrary  constants,  viz.,  the 
ratios  of  any  three  of  the  arbitrary  constants  a,  /?,  y,  d  to  the  fourth 
one.  Let  r  and  S  denote  two  corresponding  values  of  t  and  x ;  we 
have  obviously  in  the  region  of  these  values  a  relation  of  the  form 

t  —  r-  a,{x  -  ^)  +  a^x  -  5)'  +  .  .  .  , 

in  which,  when   r  or  ^  become  infinite,  we  must  replace  /  —  r  or 

jf  —  ^  by  —  or  — .     This  value  of  /  —  r  being  substituted  in  an  inte- 

t        X  ° 

gral  function  of  /  —  r  will  obviously  give  an  integral  function  of 
X  —  S.  Again,  if  it  be  substituted  in  a  regular  function  of  ^  —  r, 
such  as 

(^-^n^o+  T^  log(/-r)+  .  .  .  +  7,Iog^  {t-r)\ 

where  7^„  ,  7",  ,  .  .  .  ,  7"a  are  integral  functions  of  t  —  t,  we  shall 
obviously  have  a  regular  function  of  ;ir  —  ^  of  the  form 

{x-$yiX,-^X,\og{x-$)-\-.  ,  .  J^X,\og'{x-S)l 


EQUATIONS   OF  SECOND   ORDER— THREE    CRITICAL  POINTS.     15/ 

where  X^,  X^^  .  .  .  ,  X^  are  integral  functions  oi  x  —  $.  It  follows 
then  from  what  has  been  shown  in  Chapter  V  that  the  transformed 
equation  in  z  and  x  admits  as  critical  points  the  three  points  B,^,  B,^, 
B,^  corresponding  to  /, ,  t^,  t^,  and  that  its  integrals  are  all  regular  in 
the  regions  of  these  points.  Since  (7)  involves  three  perfectly  arbi- 
trary constants,  we  can  choose  for  the  new  critical  points  any  three 
points  we  please:  suppose  we  choose  the  points  o,  i,  00.  This  can 
be  done  in  six  different  ways,  as  shown  in  the  following  table,  t^ , 
/'^j /j  correspond  to  ^,  ,  B^,  B^,  and  by  a  proper  determination  of 
the  coefficients  a,  ft,  y^  d  (or  rather  their  ratios)  we  can  make  any 
three  values  of  B  we  choose  correspond  to  the  three  given  values 
of  ^r  we  have  then 


Denote  by  (A,  A'),  (/<,  /<'),  (r,  y')  the  roots  of  the  indicial  equations 
corresponding  to  the  critical  points  o,  i,  00 ;  then  if  the  differences 
X  —  \',jx— }j^,v—  v'  are  not  integers,  the  general  forms  of  the 
integrals  in  the  region  of  these  points  will  be: 


t. 

u 

h 

s. 

^. 

^3 

0 

I 

00 

0 

00 

I 

I 

0 

CO 

I 

00 

0 

00 

0 

I 

00 

I 

0 

(«) 


For x  =  o,      cx^U-\-  c'x^' U'\ 
Yor  X  —  I,      c{x  —  lyV -\-  c'{x  — 

For;r=co,    c~W  +  c'  —  W; 

X'^  I  ^r 


lyV; 


where  c  and  c'  are  arbitrary  constants,  [/  and  U'  are  integral  func- 
tions of  X,  Fand  V  are  integral  functions  of  ;ir  —  i,  and  IV  and  W 


are  integral  functions  of  — ;  that  is. 


158  LINEAR  DIFFERENTIAL   EQUATIONS. 

V   =  D,-\-Dix-  I)  +  Dlx-if  +  .  .  .  ; 

v  =  d:-\-  d:{x  -  I)  +d:{x  _  i)^  + . . . ; 

The  differential  equation  P  =  o  has  now  been  transformed  into  one 
with  only  the  two  finite  critical  points  x  =:  o  and  x  :=  i;  supposing 
then  A.  and  jn  not  equal  zero,  the  transformation  (4)  becomes 

(8)  2  =  x\x  —  i)y, 

and  the  corresponding  equation  between  j/  and  x  has  for  integrals: 
In  the  region  of  ;ir  =  o,       ^t/",  -|-  c'x'''-  ^U^\ 
In  the  region  of  ;tr  =1,       cV^-\-  c\x—  ly-'^V/; 


W)  i 


In  the  region  of  ;r  =  00,     c    .  ,     ,     W,-\-c'   ,1     ,   ,  W'' 


where  c  and  c'  are  arbitrary  constants,  and 

U,=  {i-x)->^U,        u:={i-x)--U\ 

V,  =  (-  lyx  -  ^  V,     v/  =  (-  lyx  -  ^  v, 

W,=  [~  -  i)"  V       W:=  (^  -  i)~'  W; 

where  {U^  ,  U/),  (F, ,  F/),  {W^  ,    W/)  are  respectively  developable 

in  positive  integral  powers  of  x,  x  —  i,  and  — . 

The  new  equation,  say  //"  =  o,  in  j/  and  x  has  thus  all  of  its  inte- 
grals regular,  and  its  indicial  equations  relative  to  the  critical  points 
X  =:  o  and  x  =  i  have  for  roots 

X  =  O]     r^  =  o,     r,  =  A'  —  A. 
X  =  I  ;     r,  =  o,     ^2  =  yw'—  j^. 


I 


EQUATIONS  OF  SECOND   ORDER— THREE    CRITICAL   POINTS.     159 

There  exist  four  ways  in  which  this  result  can  be  arrived  at ;  because 
we  can  take  for  A,  in  (8),  either  of  the  roots  of  the  indicial  equation 
corresponding  to  ;r  =  o,  and  for  fx  either  of  the  roots  correspond- 
ing to  ^  —  I.  Now,  unless  these  indicial  equations  have  A  and  A', 
yu  and  }x' ,  as  conjugate  imaginary  roots,  there  will  always  "be  one  of 
the  roots  A'—  A,  }ji! —  yw,  which  will  have  the  real  part  positive  for 
both.     The  equation  H  =  o  is  now  of  the  form 

(d)  ^>  PX^)      dy  Pjx) 

^^^  dx'  ^  x{x  -I)  dx^  x\x  -  i)=  -^ 

where  (Chapter  V)  P^x)  and  P^x)  are  polynomials  of  the  degrees 
I  and  2  respectively.  The  indicial  equations  relative  to  the  critical 
points  o  and  i  are  respectively 

^     ^  \r{r-Y)^rPXX)-\-Pli)  =  o. 

As  each  of  these  has  zero  for  a  root,  it  follows  that  P^x).  must 
vanish  for  x  ■=  o  and  for  ;ir  =  i,  and  consequently  is  divisible  by 
x{x  —  i) ;  as  P^^^  is  of  the  degree  2,  this  division  can  only  have  a 
constant  as  a  quotient,  and  the  transformed  differential  equation  is 

(II^  d^y        Ax  -^  B  dy  C  _^ 

dx"^        x{x  —  i)  dx         x{x  —  i) 

a  particular  case  of  equation  (6).  (By  Fuchs's  theorem  we  might  at 
once  have  written  the  differential  equation  in  this  form.)  The  three 
constants  A,  B,  C  are  perfectly  arbitrary,  and  may  be  replaced  by 
the  following, 

A  =  a-\-  ft  +  \,     B  =  —  y,     C  =  aft, 

so  that  (ii)  becomes 

^     ^  dx'~^  x{x  -  I)  dx^  x{x  -  i)  -^ 

1  the  well-known  differential  equation  for  the  hypergeometric  series. 
If  in  (6)  we  assumed  only  two  critical  points,  o  and  i,  then  we  should 


i6o 


LINEAR  DIFFERENTIAL  EQUATIONS. 


at  once  have  (i  i)  for  p  —  i  =  i,  p  —  2  =  o,  ^  =  x{x  —  i).     [An 
interesting  property  of  (12)  arises  from  differentiation,  viz., 

or,if^  =  ^, 

(13)  .ir(l-A-)0  +  [r+I-(«+I+^+l  +  I>]^ 

which  is  of  the  same  form  as  (12),  and  differs  from  it  only  in  having 

a  -\-  \,  (i-\-\,  and  y  -\-  \  '\n  place  of  a,  (3,  and  y.     It  is  obvious  that 

d"'y 
when  —, —  =  1]  we  shall  have 

(14)      ^(i-^')£^^+[r  +  ^«  +  («  +  '«  +  ^  +  ^«+0]^ 

—  (a  -|-  7n){^  -\-m)r^  =  Oy 

differing  from  (12)  by  having  a  -\-  jn,  ^  -\-  m,  and  y-\-  m  in  place  of 
a,  /3,  and  y.]  ,  ; 

In  order  to  get  the  indicial  equation  corresponding  to  the  point    ] 

:tr  =  00  in  (12),  transform  by  writing  ;ir  =  -  ;   we  have  then 


The  three  indicial  equations  and  their  roots  are  now,  from  (12) 
and  (15): 

r{r  —  j)  -\-  yr  :=  o, 


(16) 


For  X  =:  o, 
^  For  X  =  I, 
For;ir=  00, 


r,  =  o,     r^=  I  —  y; 

r{r  —  i)  —  r{y  —  a  —  ^  —  l)  =  O, 
r^  =  O,     r^  =^  y  —  a  —  /3; 

r{r  —  \)  —  r{a  -^  (3  —  i)  -{-  aft  =  O, 


^  =  or,     r^  =  ft. 


I 


EQUATIONS  OF  SECOND   ORDER— THREE    CRITICAL   POINTS.      l6l 

In  the  region  of  ;i;  =  o  we  have  corresponding  to  the  root  ;',  =  o  an 
integral  of  the  form 

Ji  =  C  +  C,x  +  C,^'  +  .  .  .  ; 

substituting  this  in  (12)  and  equating  to  zero  the  coefficient  of  x^, 
we  have 

\k{k  -  I)  +  ^(a  +  y5  +  I)  +  ocfi'XC,  =  \k{k  +  I)  +  y{k  +  I)]G^., 
giving 

and  so,  making  the  arbitrary  constant  C^  equal  to  unity,  we  have  for 
j/j  the  value 

(18)  ^.=    I+^JLA;.+^(^+i)^^_±_0,^   +   ...; 

\  .  Y  I  .2  .  y{y-\-l) 

or,  employing  the  usual  notation  for  hypergeometric  series. 

We  have  seen  above  that  there  are  six  ways  of  transforming  the 
differential  equation  in  question  into  an  analogous  one  when  the 
independent  variable  is  changed  by  the  relation 

(19)  ^=  ~~, 

yu  -\-  6 

and  corresponding  to  each  of  these  there  are  four  transformations  of 
the  dependent  variable  of  the  form 

(20)  y  =  x^{x  —  lys; 

where  \  and  yw  are  respectively  roots  of  the  indicial  equations  corre- 
sponding to  ;r  =  o  and  x  ^=  i. 


1 62 


LINEAR  DIFFERENTIAL   EQUATIONS. 


The  critical  points  of  (12)  are  o,  i,  00, and  the  six  different  forms 
of  (19)  are  given  by  the  following  table : 


(21) 


X  = 

0, 

I, 

00 

X   :=    U 

Ji  = 

0, 

I, 

00 

U 

u  = 

0, 

00, 

I 

U  —   I 

Jf   =    I    —  tl 

11  = 

I, 

0, 

00 

71  —  I 
f   — 

u  = 

I, 

00, 

0 

U 

I 
Y  — 

11  = 

00, 

0, 

I 

I  —  U 

I 

u  = 

00, 

I, 

0 

tc 

As   an   illustration   of   these  transformations  take   the  case   of 
u  —  I 


X  = 


The  critical   points  corresponding  to  the  original   ones 


x=:o,  xz=\,x  =  o^  are  respectively  z/  =  i,  z^  =  00,  z/  =  o,  and  the 
roots  of  the  indicial  equations  are: 


(22) 


These  values  are  written  at    once  by  aid  of  what  precedes.     The 
transformed  equation — for  which  we  have,  however,  no  use — is 


For  M  =     I, 

r,  =  0, 

r,  =    I  - 

-  r; 

For  u  =   CO, 

r,  =  0, 

r,  =  r- 

-  a 

-ft\ 

For  u  =    0, 

r,  =  a, 

r.  =  13. 

(o^\     ^>  ,  ^+/?-i-(^+/?-r-i>  dy_ aj_ 

^^^       du'  ^  u{ti  -  I)  du        u\7i  -  ly 


Make  now  the  transformation 

(24)  y  =  u^{i  —uYs, 


\ 


EQUATIONS  OF  SECOND   ORDER— THREE    CRITICAL   POINTS.     163 

*l  and  /i  being  roots  respectively  of  the  indicial  equations  corre- 
sponding to  M  =  o  and  ?/  =  i.  Referring  to  the  formulae  {a),  we 
have 

y\.  =  or  or  /?,     //  —  o  or  I  —  )/. 

Suppose  now  we  choose 

\  ^  a,     ;u=l  —  y\ 

the  formula  of  transformation  is  then 

{25)  y  —  ii\\  —  iif  -  yz. 

From  [a')  we  have  at  once,  for  the  roots  of  the  indicial  equations 
belonging  to  the  {z,  u)  equation, 

«  =     O ;         r,  =    o,  r^-=  fi  —  a\ 

(26)  ^  z^  =     I  ;  r,  =     O,  r,  =  ;^  —  I  ; 

?^  =    CO  ;  r^  ^=.  a  A^  \  —  y,  r,  =  I  —  ytf. 

The  (j,  w)  equation  is 

{27)     ^  +  (^-^-0-(A+r-^-3>  dz_  _  {fi-\){a-y^\)^_^^ 
dii"  iciji  —  i)  (in  u{u  —  i) 

giving  at  once  the  roots  in  equations  (26).     Write 

a'  =  a  -  y  +  I,       /3'  —  l  —  j3,      y'  =  a  —  j3  -{-  I  ; 

then  this  equation  has  in  the  region  of  //  =  o  the  integral 

2  =  F{a',  13',  y',  u). 

Transforming  back  to  the  variables  y  and  x,  we  have,  on  neglecting 
a  constant  factor  which  is  merely  a  power  of  —  i, 

(28)     J  =  ^-'^(l-iy~""V(«-;.+  I,    I-A    «-/J+l,  _-l-). 

Proceeding  in  this  way,  it  is  clear  that  we  can  form  24  particular 
integrals  of  (12). 


164  LINEAR  DIFFERENTIAL   EQUATIONS. 

A  full  discussion  of  these  functions  is  contained  in  the  follow- 
ing chapter,  which  is  a  translation  of  a  Thesis  by  M.  E.  Goursat  ; 
the  reader  is  also  referred  to  another  memoir  by  the  same  author 
in  the  Annales  de  V Ecole  Normale  for  1883.  In  the  following- 
table  of  the  24  functions  we    must  make  some   supposition   as   to 

the  values  of  the  different  powers  of  x,  i  —  x,  \ which   appear 

as  factors:  suppose  we  choose  the  values  such  that  the  different 
powers  X,  \  — x,  \ shall  reduce  to  unity  when  we  make  respec- 
tively X  ^  I,  ;ir  =  o,  ;f  =  00. 

I.      F{a,  /?,   y,  x).  o  '        --1 

II.      (I    -  xy  ---^F{y  -a,    y  -  /3,    y,   x). 

III.  (I  -  x)-'^F{^a,  y-  ft,  y,  ~^  . 

IV.  (I   -  x)-^F[ft,  y-a,y,  ^-)  . 

V.      X'  -  yF{a  —  ;/  +  I ,   /3  —  y  -\-  \,   2  —  y,  x). 
VI.     X' -y{i  —  x)y-'^-PF{i  —  a,   I  —  p,  2  —  y,  x). 

VII.     x'-y{l  —  x)y-'^-'F{a  —  y  -\-  I,  I  —  ft,   2  —  y,  — 3— )•• 

X     \ 


VIII.  x'-y{l —x)y-P-'F[ft- y-\-  I,  1  —  a,  2  ~  y,      _ 

IX.  F(a,  ft,   a  -}-  ft  —  y-\-  I,    I  ~  x). 

X.  x'-yF{a  —  y-ir  I,  a-\- ft  —  y-{- I,   I —^). 

XI.  x-'^F[a,  a-  y-^l,   a-\- ft-  y-^  I,^^^-). 

XII.  x-^F[ft,   ft-y^l,   a^  ft-y+  I,   --^'-^ 

XIII.  (I  -  x)y---^F{y  -a,  y-  ft,  y-  a  -  ft-\-i,    I  —  x). 

XIV.  (i  -  x)y-''-^x'-yF{i-a,  i-ft,  y  -  a  —  ft -\-  i,    j  -  x). 


EQUATIONS   OF  SECOND    ORDER— THREE    CRITICAL   POINTS.      1 65 
XV.      (l  —  xy-"^-^  x^-ipyi  —  a,  y  —  a,y  —  a—  ft-\^l,  1. 

XVI.     (i  -x)y-'^-^x^-yF\^i  ~  j3,  r-  /3,  y-a-  /3-^i,  ^^^j. 

'      XVII.     x-'^F[a,   a-y-^l,   a- /3^  I,  -]. 

XVIII.    x-{i  _iy"""V(i-A  r-A  ^  -/?+!,  '-)' 

XIX.     x-^i-^y^FL  y-^,  a-  ^+1,   -^j.        ^ 


xi  \       '         •  I  —  X 


XX.     x-\\--^  F\a—y^\,\—ft,  a  -fi^\. 


XI  \  '      '  ■       '  I  —  X. 


y.     XXI.  ;r-^/^(/?,  /?  -  ;/+  I,  /? -«+  I,  -). 

XXII.  ;r-^(l-^j       "       /^(l  -a-,   ;K-  o',   /5— «r+  I,  -]. 

XXIII.  .^-^(l-^)"V(/?,   ;.-^,  /?_^+I,   _1-). 

XXIV.  x-\l--j  F[f3-y-\-i,i-a,^-a  +  i,- 


The  general  condition  for  the  convergence  of  the  series  F{a,  ft,  y,  $) 
is  mod.  ^  <  I  ;  from  this  we  can  see  at  .once  what  are  the  regions  of 
convergence  for  the  above  series  of  functions,  viz. — (i) :  for  the  argu- 
ment X  the   region  of  convergence   is   the  interior  of  the  circle  of 

radius  unity,  whose  centre  is  at  .«•  =  O ;  (2) :  for  the  argument  -  the 

region  of  convergence  is  the  entire  plane  outside  of  this  circle;  (3): 
for  the  argument  i  —  x  the  region  of  convergence  is  the  interior  of 
the  circle  whose  centre  is  at  the  point  x  =  i  and  whose  radius  is 

unity;  (4):  for  the  argument the  region  of  convergence  is  all 


1 66  LINEAR  DIFFERENTIAL  EQUATIONS. 

of  the  plane  outside  of  this  circle ;  (5) :  for  the  argument sup- 
pose X  ^=^  B,  -\-  ij],  and  make  ^  =  |-,  then 

mod. =  mod.  — - — ; — :-  =  i, 

and  so  the  region  of  convergence  is  all  of  the  plane  to  the  left  of 

^  —  I 
the  line  ^  =  ^;  (6) :  for the  required  region  is  obviously  all  of 

the  plane  to  the  right  of  the  same  line. 

The  dissection  of  the  plane  which  is  necessary  in  order  that  the 
above  functions  may  remain  uniform  while  the  variable  travels  all 
over  the  plane  without  crossing  any  of  the  cuts  is  obviously  affected 
by  drawing  a  cut  from  o  to  —  00  ,  and  another  from  -j-  i  to  -(-  00, 

It  is  now  easy  to  see  that  the  above  24  functions  divide  into  six 
groups  of  four  each.  In  the  region  of  the  point  x  ^=  o  the  indicial 
equation  has  for  roots  o  and  i  —  y  ;  now  the  functions  I  to  IV 
inclusive  are  uniform  and  convergent  in  the  region  of  the  point 
X  =  o,  and  they  reduce  to  unity  for  x  =  o;  there  can,  however,  be 
but  one  integral  in  this  region  possessing  this  property,  viz.  the 
integral 

fi  =  ^{^,  /3>  r>  ^) 

belonging  to  the  exponent  zero ;  therefore  each  of  the  integrals 
I,  II,  III,  IV  represents  the  same  function.  Again,  the  functions 
V  to  VIII  inclusive  are  the  products  oi  x^  -y  into  expressions  which 
are  uniform  and  convergent  for  the  same  region  x  =^  o,  and  which 
reduce  to  unity  for  ;f  =  o  ;  they  represent  therefore  the  one  regular 
integral,  say  y^,  corresponding  to  the  root  i  —  y  oi  the  indicial 
equation  for  x  =^  o.  The  expressions  IX  to  XII,  and  XIII  to  XVI, 
represent  in  the  same  way  the  two  regular  integrals,  sayjj/j  and  jj/, , 
corresponding  to  the  point  ;ir  =  i ;  and  XVII  to  XX,  and  XXI  to 
XXIV,  represent  the  two  regular  integrals,  say  y^  and  j'„ ,  belonging 
to  the  point  ;ir  =  00. 

It  will  be  convenient  to  insert  here  a  direct  determination  of  the 
integrals  of  (12)  which  belong  to  the  critical  points  o,  i,  and  00 
respectively,   and  to   show  what   conditions  must   in   each   case  be 


I 


EQUATIONS  OF  SECOND   ORDER— THREE    CRITICAL   POINTS.     1 6/ 

satisfied  by  the  constants  a,  (3,  y,  in  order  that  the  integrals  may  be 
of  the  forms  sought.     Writing  equation  (12)  in  the  form 


(29)  ix^  _  ^)  g_  _  [^  _  (^+  ;?+  i)^]  g  -f  a^j;  =  o, 


we  have  as  above,  for  the  roots  of  the  indicial  equations: 

For  X  =    o,     r^  =  o,     7-^  =  i   —  y; 
For  X  =    I,     r,  =  0,     r^  =  y  —  a  —  ^  ; 
For  ;ir  =  CO,     r^  =  a,     r^  =/3. 

When  y  is  not  a  negative  integer  we  have  in  the  region  of  x  =  O  the 
regular  integral 

where  F{a,  /?,  y,  x)  is  a  convergent  series  inside  the  circle  whose 
centre  is  ;«;  =  o  and  whose  radius  is  unity.  Inside  this  same  circle 
there  must  exist  another  integral  belonging  to  the  exponent  i  —  y; 
to  find  it,  write 

y  =  x'-yYy 
and  we  have  for  V  the  equation 

(30)  {x'  -  x)^-^  -  [2  -  y  -  {a-\-  ^  -  2y  +  3)^]  — 

-  (^  -  r+  i){/3-y+  i)F=o; 

which  differs  from  (29)  only  by  having  a,  ft,  y  replaced  hy  a  —y-\-i, 
/3  —  y  -\-  I,  2  —  y.     When  2  —  y  is  not  a  negative  integer,  equation 

(30)  admits  the  solution 

(31)  Y  =  J^(^a  -  y  -{-  I,  ft  -  y  -{-  I,  2  -  y,  x), 


1 68  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  consequently  (29)  has  the  solution 

(32)  y^  =  x'-yF{a-  y^j,^-y-{-i,2-y,x). 

It  follows  then  that  if  y  is  not  an  integer,  the  general  integral  of  (29) 
is,  in  the  region  of  ;r  =  o, 

i 

(33)       •  j'  =  C}\  +  cy,, 

where  C  and  C  are  arbitrary  constants.     For  the  region  of  .«■  =  i 
make  x  =  i  —  6  ;  equation  (29)  becomes  now 


72  J 


which  differs  from  (29)  only  by  the  change  of  y  into  ex  -\-  /3  —  y  -\-  i. 
If  then  a  -^  f3  —  y  -\-  I  is  not  an  integer,  (29)  admits  in  the  region 
of  the  point  x  =  1  the  solutions 

.        (  J',  =  F{a,  /3,  a-\-0-y-\-l,  i-x); 

\  y,  =  xy-''-  ^F{y  —  (5,  y  —  a,  I  -\- y  —  a  —  /3,  I  —  x)  ; 

and  the  complete  solution  in  the  region  of  this  point  will  be 

where  C  and  C  are  arbitrary  constants.     For  the  region  of  a-  =  co 

it  is  only  necessary  to  write  x  =  —  ,  and  find  the  indicial  equation  for 

i  =  O.  We  find  readily  that  ii  a  —  ^  -\-  i  is  neither  zero  nor  a 
negative  integer, 

(36)  ■y^  =  x--F[a,  a-  y-\-i,  a-  ^^i,^) 


EQUATIONS  OF  SECOND    ORDER— THREE    CRITICAL   POINTS.       1 69 

is  an  integral  oi  (29) ;  and  so  if  /?  —  a  +  I  is  neither  zero  nor  a  nega- 
tive integer, 

(36)  y.  =  x-^F[p,  /J  _  ^  +  I,  ^  -  «  +  I,  L 

These  are  the  six  integrals  spoken  of  above ;  as  to  their  regions  of 
convergence,  it  is  clear  that  for  the  pair  (7, ,  y^  we  have  the  circle  of 
radius  unity  whose  centre  is  at  the  origin  ;  for  the  pair  (j/, ,  y^  we 
have  the  entire  plane  outside  of  this  circle  ;  and  for  (j/3,  y^)  the  circle 
whose  centre  is  at  ;ir  =  i  and  whose  radius  is  unity. 

We  see  now  that  the  regions  of  ;f  =  o  and  x  =  i  have  an  area  in 
common,  as  have  also  the  regions  oi  x  =  i  and  x  =  cx>.  li  i  —  y 
and  y  —  a  —  /3  have  their  real  parts  positive  or  zero,  the  series 
jj/j ,  .  .  ,  ,  j/g  will  be  convergent  at  each  point  on  their  respective 
circles  (a  known  property  of  the  convergence  of  these  series  which 
need  not  be  gone  into  here);  it  follows  therefore  that  in  the  area 
common,  say,  to  ^  =  o  and  ;tr  =  i  the  functions  j/g  and  j^  are  linearly 
expressible  in  terms  of  y^  and  y^ .  If  now  in  this  common  area  we 
have  expressed  y^  and  y^  in  terms  of  j,  and  y^ ,  we  can  by  a  known 
process  in  the  theory  of  functions  obtain  uniform  and  convergent 
developments  for  y^  and  y^  in  the  whole  region  of  x  =  o.  Similar 
remarks  apply  to  the  linear  relations  connecting  the  integrals  j/^  and 
jj/g  with  j'3  and  y^ ,  and  consequently  with  y^  and  y^ .  Following 
Jordan,  write  for  subsequent  convenience,  instead  oi  y^,  .  .  .  ,y^, 

^i^i,    ^.7,,    ^3^3.    ^4^4,    ^.y.,    ^6^6 ; 

then  it  is  clear  that  we  must  have  the  relations 

.         j  ^3^3  =  Ac,y^  +  Bc,y, ;  (  c,y,  =  Ec,y,  +  Fc^y^ ; 

^^^^    \  c.y.  =  Cc,y,  +  Dc^y, ;  ^^  M  c,y,  =  Gc,y,  +Hc,y,; 

where  the  constants  A,  .  .  .  ,  Hhave  now  to  be  determined. 

Suppose  X  to  describe  a  circle  of  radius  unity  around  the  origin  ; 
the  integral  y^  will  not  change,  but  y^,  y^,  y^  will  be  multiplied 
respectively  by  e^"'^"^  -y\  ^-='^^'",  ^-^"'^  ,  and  equations  (38)  become 

(  e-  -- c,y,  =  Be,  y,  +  Fe^^^' ('  -  v)  c,  y, ; 


lyo 


LINEAR  DIFFERENTIAL  EQUATIONS. 


Now  in  equations  (37)  make  ;tr  =  o;  also  in  (37),  (38),  (39)  make 
or  =  I,  and  we  have  the  following  eight  equations  for  the  determina- 
tion of  the  constants  A,  .  .  .  ,  H\ 

^"3(^3)1  =  A<^A}\\  +  Bcly,\  ;     c,{y,\  =  Cc,{yX  +  ^^,(  J,),  I 

(40)   i  ^,( ja),  =  £^Xj'X  +  ^-^".(j^.). ;   ^6( jj,  =  <^^.( J.).  +  ^^.(7,). ; 


The  values  of  (jj/,)„ , 

(7.)o  =  I ; 

(j''3)« 


(41) 


,  ( r^),  are  known  to  be 
Mo  =  o ; 


r,(a+/?-;K-fi)r(i-r) 


r{/j-y+i)r{a-y-\-i) 


(v\-  r(r-Q--/g+i)r(i-r) 


,       _r{r)r(r-a-(3)^ 
\j  J.  —  j-(^  _  a)ri,r  -  /i) ' 

(jO.  =  I ; 

_   r(cr-^+i)r(;K-n--/?)  . 


{y.\ 


_r(2  -yWir-  a  -  /3) 


{yX 


rn-  a)r(i  -  /3) 

{y.\  =  o  ; 

ri/3-a-\-i)r(r^a-^) 


r{i-ii)r{r-(5) 


F-';   W.= 


r(i  -  a)r(;K  -  or) 


These  values  introduced  in  (40)  give  us  the  means  of  determining 
the  constants  A,  .  .  .  ,  H  \n  terms  of  the  constants  c^,  .  .  .  ,  c^^ 
exponentials  and  /^-functions.  A  choice  of  values  must  now  be 
made  for  the  constants  c^,  .  .  .  ,  c^;  \{  we  make  them  each  equal  to 
unity,  then  A,  .  .  .  ,  H  are  determined  as  functions  of  the  expo- 
nentials and  the  /^-functions.  A  still  simpler  result  is  obtained  by 
writing 


(42)     \ 


r{a)T{y  —  a) 
C. 


i\2-r) 
r(i  -  /?)r(r  -  a) 


r(«  +  /?-r+0'    '     r{y  -a-  ft+i)' 


_7>)r(i-/?). 
^'-r(a-^+i)' 


c.  = 


r{^-y-\-l)r{y-a) 
r{/3-a-i-i) 


EQUATIONS  OF  SECOND    ORDER— THREE    CRITICAL  POINTS.     I/I 

Equations  (40)  are  now 


(43) 


r(^)r(i  -  ;.) 

r{a  -  /  +  I) 
r(i  -  y)r{y  -  a) 


=  A 


=  c 


r{a)r{r  -  a) 

r{a)r{y  —  a) 


r{i  -a)  -        r{y) 

r{a)r{l3-yJrl)  ^  ^  r{y-a-  f3)r{a) 


Tla-^  fi-y+  I) 


0=  c 


r{y-a-f3)r{^-y+l) 
r{i  -  a) 

r(y-a  -  fi)r{a) 

r{r  -  /i) 

riy-a-/3)r{l3—yJ^l) 


r(i  -  a) 

r(y-a-  p)r{a)  ^       r(y-a-  /3)r{a) 
r{y  -  /?)  r{y  -  /3) 

r{y-a-fi)r{/3-y+i) 


+  ^ 


r(i  -  a) 


If  we  divide  each  of  these  equations  by  the  product  of  the  /^-func- 
tions which  appear  in  each  numerator  and  then  employ  the  relation 


TT 


we  have  finally 

sin  {y  —  a)TC  =  A  sin  yn  ;     sin  an  =.  C  sin  yn  ; 
sin  {y  —  a  —  I3)7t  =  A  sin  {y  —  f3)7t  -\-  B  sin  an  ; 
O  =  C  sin  {y  —  §)n  +  D  sin  an ; 
(44)       \  sin  {y  —  0)n  =  E  sin  {y  —  fi)n  +  F  sin  an  ; 

sin  an  =  G  sin  {y  —  fd)n  +  H  sin  an ; 
<?-^'^'»sin  {y  —  p)n  =  E  sin  {y  —  ft)n  -\-  /^^='^'(^-v)sin  an\. 
^-^'^'^sin  o'TT  =  G  sin  (;(/  —  /?)7r  +  i/^^'^'d  -  v)  sin  ar;r.- 


^T2  LINEAR   DIFFERENTIAL  EQUATIONS. 

These  give  A,  .  .  .  ,  H  \n  terms  of  sines  and  exponentials  only,  viz. : 

.  sin  {j  —  oi)7i 

/i  — ; 


<45) 


B 


sin  yiT 
I 


sin  (X7t    _ 


sm{y — a—^)7t- 


sin  {y—a)7i  sin  {y~0)i: 
sin  yn 


']■■ 


^  sin  an 


sin  yn 


„  __       sin  an  sin  {j  —  (i)n 
sin  an  sin  yn 


F 


G 


H 


^,2Ttl(  I    -   Vj     J 

c-  ="'■"  —  I         sin  (/?  —  y)7r 


^.27r/(i  -  V)    J 

I 


Sin  an 
sin  an 


^-  27r/|8   —    J 


sin  {y  —  P)n       sin  {y  —  fi)n       e^'"'^^  -  7)  _  i  ' 


These  forms  have  been  taken  from  Jordan ;  they  are  to  be  found  in 
a  number  of  places,  e.g.,  Forsyth's  Differential  Equations  (here  in  a 
somewhat  modified  form)  and  Goursat's  Thesis,  above  referred  to. 

If  y  (and  therefore  also  i  —  ;>/)  is  an  integer,  then  one  of  the 
integrals  in  the  region  of  ;r  =  o  niay  contain  a  logarithm;  so  if 
y  —  a  —  ft  is  an  integer,  or  if  or  —  ^  is  an  integer,  a  logarithm  may 
appear  in  one  of  the  integrals  belonging  to  ;r  =  i  and  to  ;ir  =  00.  It 
has  been  shown  in  general  that  logarithms  enter  in  such  cases  as 
these  ;  they  may,  however,  fall  out  even  when  these  conditions  are 
satisfied.    This  question  will  be  entered  into  later. 

A  single  illustration  of  the  entrance  of  logarithms  will  only  be 
given.  We  have  already  seen  that  if  the  two  roots  of  an  indicial 
equation  are  equal,  then  a  logarithm  must  necessarily  appear.  Sup- 
pose now  that  we  have  y-=\,  and  consider  the  region  of  ;f  =  o. 
The  roots  of  the  indicial  equation  are  now  each  zero,  and  the  dif- 
ferential equation  itself  is 


EQUATIONS  OF  SECOND    ORDER— THREE    CRITICAL   POINTS.      1 7 3', 

(46)  ^(;,_l)^+[(^  +  /5+l)^_l]g+^;?^    =    0 

one  integral  of  which  is 

F(a,  13,   I,  x),     =  say,  A,  +  A,x  +  A^x'  +  .  .  .  , 
when  for  brevity  we  have  written 


A, 


a{a+l)...  {a  +  k  -  l)ft  {ft -^  l)  .  .  .  {/S -\-  k   -l) 


ill 
For  convenience  write 

^  =  F{a,  ft,  I,  x); 

then  a  second  integral  of  (46)  is  of  the  form 

y  =  (P  -\-  0  log  X, 

where  0  is  a  uniform  and  continuous  function  of  x  in  the  region  of 
X  =^  O.  Substituting  this  value  of  y  in  (46),  and  equating  to  zerO' 
the  aggregate  of  terms  which  do  not  contain  the  logarithm  as  factor^ 
we  have  for  0  the  equation 

(47)  ^(^-    l)^^  +  [(a+^+l)^-l]g+^^0+2(^-   I)-^ 

+  {a  +  ft)0=O. 
Assume  now 

(48)  0  =  A, a,  +  A,a,x  +  A,a,x'  +  .  .  .  , 

and  replace  in  (47)  ^  and  0  by  their  development  in  series;  we  have, 
on  equating  to  zero  the  coefficient  of  x'', 

(49)  {a  +  k){ft  +  ^)  A,a,  -(k+iyA,  +  ,a,  +  , 

J^{a-\-ft-^2k)A,-  2{k+i)A,+,  =  0; 

giving,  since 

A  +  .  ^  {a  +  k)(ft-^k) 
A,  {k+if       ' 

/-  \  —2,1,1 

(50)  ak  +  ,  —ak  =  -j—^ —  +  ■ — — .  + 


^+  I    '    a-{-  k    '    ft-\-  k 


174  LINEAR  DIFFERENTIAL  EQUATIONS. 

From  this  last  we  have 


We  may  make  «„  =  o  without  loss  of  generality,  since  this  only 
amounts  to  subtracting  a^^  from  0;  the  series  giving  0  is  thus 
determined  by  its  general  term  Aka^^. 

It  is  not  difficult  to  see  from  the  form  of  this  general  term  that 
the  solution 

0  -(-   <?  log  X 

thus  obtained  can  be  written  in  the  form 

,,  ,    d^    ,    d^    ,    d^ 

^log. +  _  +  _  +  _; 

d^   d^    d^ 
where  -i— ,  -tt,  ,  -7—  denote  the  values  of  the  derivatives  of  the  func- 
tor  dp    dy 

tion 

^  =  F{a,  /?,  y,  x) 

when  we  have  made  y  ^=1  \. 

It  is  not  possible  to  give  any  adequate  account  of  the  many 
investigations  that  have  been  made  upon  the  hypergeometric  series, 
but  the  results  obtained  by  M.  Andre  Markoff  in  two  interesting 
notes  in  the  Mathematische  Annalen  may  be  here  stated.  In  his 
first  note*  M.  Markoff  proposes  to  find  all  the  cases  where  the 
product  of  two  values  of  j  satisfying  the  differential  equation 

^('  -^'^^^  +  t^  -  (^  +  ^+  ^W  ^  -  ^^-^  ==  ° 
*  Math.  Annalen,  vol.  xxviii.  p.  586. 


EQUATIONS  OF  SECOND   ORDER— MARKOFF' S    THEOREMS.       1/5 

I 

I  reduces  to  an  integral  function  of  x.     Writing  z  ^=  y^y^,  the  differ- 
ential equation  of  the  third  order  satisfied  by  z  is  found  to  be 

dz 
+  (ex-"  J^dx-^e)  —  -^{fx-^  g)z  =  o; 


where 

«=-(«  +  /?+!), 

b  =  y, 

c  =  2a'  +  Saj3  +  2/?'  -|-  3^^  +  3y5  +  I, 

d==  -  2y{2a  +  2/3  +  I)  -  4«/?, 

e  =  2y''  —  y, 

f  =  4«/?(^  +  /?). 

^  =   -  2a'/?(2;K  -   l). 

It  remains  now  to  find  all  the  cases  where  this  last  differential 
equation  admits  the  solution 

z  =  integral  function  of  x. 

If  n  denote  the  degree  of  the  function,  the  conditions  are  found  to 
be  as  follows: 

(A)     n  even  : 
<i)  «^  =  -  2. 

i{2)^=--, 

/  V      ■ ,    ^                       I            I            3                         2n  —  I 
l{7,)   a-{-^  =  -n,  y  =  -,      -  -,      - -,     ..., ; 


1 76  LINEAR  DIFFERENTIAL  EQUATIONS. 

(B)     «  odd : 

n                I            13                         ?z  —  2 
(:)«=--,     y=-,     --,     --, —, 


n  —  I 
A     /?-  I,     .  .  .,     /? —, 


n 

I             I 

r=2'    -2' 

3 

2' 

•  • 

> 

n  —  2 

2     ' 

?i  —  I 

^, 

O"   - 

-  I, 

..   .,      a 

(3)   «+/?  = 

-  «.    r  =  2' 

I 
~  2' 

3 

■  2' 

.  .  . 

71  —  2 

2       ' 

n 

2n  —  I 

In  his    second  note  *   (closely  allied  to   certain    investigations  by 
Schwarz)  Markoff  seeks  to  find  all  the  cases  where  the  same  differ-    | 
ential  equation  of  the  second  order  admits  an  integral  of  the  form 

where  X,  V,  and  Z  are  rational  integral  functions  of  x.     The  con- 
ditions found  in  this  case  are  as  follows  (;z  an  integer): 

{i)    a  -\-  /3  =  —  n,     r  =  h     —  h     -  h     •   •  •  '     —  «  +  2, 
{2)    a  -^  /3  =  n,     r  =  ^,     I,     .  .  .  ,     n  —  ^, 

{3)  ^  +  ft  -  ^r  =  -  ^^>   r  =  §,   h    ■  ■  ■ '    ^^  -  h 

{4)    a  -{-  /3  —  2y  =  n,     r  =  h     —  i,     —  f ,     •  •  •  ,     —  «  +  i, 


2a  —  y  ) 
(5)         or        V=—n,     a-\-/3  —  y=—i,      —  f ,     ...,      —  « +  i, 
2^-y) 


2a  —  y 
or 

2ft  -y 


(6)         or        \=  n,     a  -\-  ft  —  y  =  \,     |, 


Math.  Annalen,   vol.  xxix.  p.  247. 


EQUATIONS  OF  SECOND   ORDER— ABELIAN  INTEGRALS.         IJJ 
a-  f5  +  y\ 

(7)        or        \=  —  ''h    r  =  h    —  h    —  h    •  •  • »    — « +  2 ; 


a  -  /3-\-  y 
/3  —  a  -\r  y' 


(8)  or  i  =  n,     y  —  ^      ^  ^  _  i 


An  interesting  case  of  the  differential  equation  of  the  second 
order  has  been  studied  by  Dr.  Heun  in  \\iQ  Mathematische  Annalen.'^ 

If  ^j,  ^2,  .  .  .  ,  S,p  are  the  finite  critical  points  of  the  equation, 
and  if  ^{x)  =  (x  —  5',),  {x  —  S^),  .  .  .  ,  (x  —  ^p),  we  have  seen,  equa- 
tion (2),  that  the  differential  equation  can  be  put  in  the  form 

(52)  ^W  ^  +  aw  ^  +  QM7  =  o ; 

where  Qi{.r)  and    Q^{x)  are  polynomials  of  the  degrees  p  -i-  i  and 
p  —  2  respectively. 

The  conditions  f  that  the  points  S  shall  be  critical  points  deter- 
mine p  -f-  I  of  the  2p  —  I  constants  in  (52),  and  there  remain  there- 
fore (in  Q^i-r))  p  —  2  independent  constants,  which  we  will  call  the 
characteristic  parameters  of  (52).  Equation  (52)  written  out  fully  is 
of  the  form 

(53)  ^<-^)^  +  (^.^''-'  +  4-^''-^+-  •  '-^K)% 

+  {k:x^  -  ^  +  k.'xf  -  3  + . . .  +  k/)y  =  o. 

Denoting  by  {x,  i)»  a  rational  integral  function  of  x  of  degree  «,  and 
writing 

J/,  =  {x,   I)^ 

then  in  order  that  j,  may  satisfy  (53)  we  must  have 

n{n  —  i)  J^  k,n-\-  k^  =  o. 


*  "  Integration  regularer  Differentialgleichungen  zweiter  Ordnung  durch  die  Ket- 
tenbruchentwicklung  von  ganzen  Abel'schen  Integralen  dritter  Gattung."  Von  Karl 
Heun  in  Miinchen.     Math.  Annalen,  vol.  xxx.  p.  553. 

\  American  Journal  of  Mathotiatics,  vol.  x.  p.  205  et  seq. 


178  LINEAR  DIFFERENTIAL   EQUATIONS. 

This  condition  is  manifestly  not  sufficient,  as  the  characteristic 
parameters  k^ ,  k^,  .  .  .  ,  k/  must  satisfy  certain  algebraic  equations 
found  by  substituting  y^  in  (53).  These  conditions  serve  for  the 
complete  determination  of  the  characteristic  parameters.*  The 
number  of  integral  functions  j, ,  =  {x,  i)" ,  obtained  in  this  way,  is 
equal 

(n  -f  i){tt  +  2)   .   .  .    {n^p  -2)  _ 
I  .  2  ...  (p  —  2) 

From  these  rational  solutions  of  (53)  the  second  (and  transcendental) 
integral  is  obtained  by  the  known  formula 

(54)  .r.  =  7.  /  — -1 —  ^^- 

In  the  special  case  where  Q^i^)  =  -  -y-  ip{x)  Heine  has  shownf  that 
we  have 

(55)        i^)=''.l.'='J,,(^^^i^)Wr^'' 


where  Z'  is  a  rational  integral  function  of  x,  and  C, ,  C^,  .  .  .  ,  Cp  - , 
are  constants  determined  by  aid  of  a  certain  system  of  algebraic 
equations.     This  last  equation  can  be  written  in  the  form 

(56)  J/.  =  yj\  +  £> 

where  f^and  E  are  functions  of  x  satisfying  the  relation 


(57)  :y: 


dx 


'^  dx\y)_ 


"?/•«-) 


/C'll'  ) 


e 


*  Heine  :   Handbuch  der  Kugelfunctionen,  Bd.  I,  §  136. 
f  Crelle,  vol.  6i. 


EQUATIONS  OF  SECOND   ORDER— ABELIAN  INTEGRALS.         1 79 
viz.,  from  (56)  we  have 

'^^  dxKyJ  ~~    dx    '^  dx\yj' 

and  from  (54) 


<59) 


d^{yj\  _  _  e   "^^'-^^ 


dx 

dx\y^, 


7: 


■equating  the  right-hand  members  of  (58)  and  (59),  we  have  (57). 

This  result  of  Heine's  is,  of  course,  restricted  by  the  condition 
imposed  upon  Q^{x),  viz., 

Starting  from  equation  (57)  we  can,  however,  determine  Fand  E  in 
the  general  case.     Write 

(60)  y^  =  -.^T^,     V  =  I,     2,  .  .  .  ,     p. 

Since  the  polynomials  Q^-^)  and  i/-(x)  are  respectively  of  the  degrees 
p  —  I  and  p,  we  have 

(61)  91^}  =  -J^-  +  _2^-  +  .     +  _r^_ . 

Write,  again, 

(62)  {x - a:)y.{x - ^,)v. .  .  .{x- g,y<>  =  e{x) ■ 

now  in  (57)  we  can  make 

where  Z  is  a  rational  integral  function,  as  yet  to  be  determined, 
whose  degree  is  ;z  —  i,  and  (p{x)  is  an  equally  undetermined  rational 
integral  function  whose  degree  cannot  exceed  p  —  2. 


l80  LINEAR  DIFFERENTIAL  EQUATIONS. 

Equation  (56)  gives  now 


(64) 

or 

(65) 


y^ 


e{x) 


e{x) 


z, 


%) .,  _.,  %)  r<t>{x) 


■J.  =7i 


/- 


dx  —  Z. 


Denoting  now  by  W  a  transcendental  function  of  x,  we  have 

{66)  i?*"'  =  A^w  W  -  Z'«) ; 

where  Z*"*  -^  iV'"'  is  the  n^^  convergent  of  the  development  of  W  in; 
continued  fractions,  and  R^"^  is  the  corresponding  remainder.  We 
can  now  write 

if  we  can  establish  the  following  conditions : 

I.  That  W  is  developable  in  a  continued  fraction  of  the  form 


W  = 


«o  +  b,^        + 


c. 


a^  4-  b^x     -f" 


C 


a^-^b^x     +     .  . 


which,  for  a  given  value  of  n,  is  regular  up  to,  and  including,  the  n^^ 
partial  denominator. 

II.  If  there  exist  as  many  values  of  0(;tr)  as  there  are  rational 
solutions  j/j  of  the  differential  equation,  viz., 

{n  +  \){n  +  2)  .  .  .  (;?  +  p  —  2)  _ 
I  .  2  ...  (p  -  2) 

^lix) 

III.  If  the  function  7?'"'  -Tir^-  is  a  second  solution  of  the  differ- 

d{x) 

ential  equation. 


EQUATIONS  OF  SECOND   ORDER— ABELIAN  INTEGRALS.         l8l 

To  show  that  W  is  developable  in  a  continued  fraction  of  the 
form  in  (I),  it  is  necessary  first  to  show  that  W  is  developable  in  a 
convergent    series    containing    only   negative   powers  of  x.     Write 

^.  .    =/2(^);  then,  by  a  known  theorem  of  Abel's,* 
dz  „.  .    r         dx 


<'58)   ^w/zrir^oT^^-^)/; 


{x  —  z)D.{z)  ^  '^   ip  —  x)0{x') 


m=p—2    n=p-2     t  7/7-1- T  nA-l 


tn  =o       n  —  o 


where 


^       p  x"'dx         n  z"dz 

J,n,n    -   J    -0U)~'    J    Tlfs)' 


e{x)     J  n{s) 

0{x)  '^-^~-  =  b^°^  +  b^'^x  +  .  .  .  +  ^"'  -  ')a'P  -  S 

d6(v\ 
n{x)  — x^-  =  ^°^  +  ^'^^  +  .  .  .  +  f^''  -  ');trP  -  '. 

The  roots  of  the  equation  0{x)  =  o  are  ^, ,  S^,  .  .  .  ,  ^p.  In  (68) 
integrate  from  one  of  these  roots,  say  Sy,  to  the  following,  ^^.j. , ;  we 
thus  obtain  expressions  of  the  form 


(69)    cp{x)    ^ 


hv   (^  ~  ^)-^-^(^) 


>K  =  O 


where  the  quantities  F  are  known  constants.     Multiply  now  each  of 
equations  (69)  by  the  corresponding  constant  of  the  series 

C       C  C 

and  add  the  results  ;  we  have  then 

*  Jacobi:  Gesammelte  IVerke,  Bd.  II.  pp.  125-127. 


l82 


LINEAR  DIFFERENTIAL  EQUATIONS. 


Letting  6  denote  a  constant,  we  have,  from  what  precedes,  for  4>{x). 
the  form 


(71) 


<p{x)  =  S,^d,x^6,x-'^.  .  .  +  dp_,^''-^ 


From  this  it   follows  very  easily  that  the  integral  on  the  left-hand 
side  of  (70)  is  identical  with  the  above-given  value  of  ^Fif  we  have 


(72) 


W^now  takes  the  form 


(73) 


w 


V  —  p—  I 

2    C 


-'    6{s) 


\.    (^  -  2)ij-{2) 


dz. 


From  this  equation  it  immediately  follows  that  ^is  developable  in 
a  convergent  series  containing  only  negative  powers  of  x. 

In  what  follows  we  will  assume  that  B{p)  has  the  form 


(74)  e{z)  =  {z-  a,r  (^  -  ^.T  .  .  .  (^-  -  s.Yp, 

where  A^  and  i^k  are  positive  integers  satisfying  the  condition 

Ai  <  /^A . 
Under  this  hypothesis  we  have 
(  .\      n(  \        ^^     (.r  -  a,){x  -a^)  .  .  .  {x-  a,) 


y",  ■  (<? --.X^ -^s) 


(^1—  ^p) 


\     {x  -  a^{x  -  B^)  .  .  .  {x-  g,_0 

and  consequently  the  function  W  is  a  sum  of  entire  Abqlian  inte- 
grals of  the  third  kind. 


EQUATIONS  OF  SECOND    ORDER— ABELIAN  INTEGRALS.         1 83 

We  proceed  now  to  the  complete  determination  of  the  function 
<t>{x).  (One  coefficient  only  in  <p{x)  will  remain  indeterminate.) 
From  equation  (56)  and  the  following  ones  we  have 

Dropping  the  x  when  it  is  not  necessary  to  write  it,  we  have  from 
these  last  equations 


t 


J^in 


»'     =yjp^-lz-^, 


or 


giving 


or 


{7^) 


tp     i?(«'  /»0   ,         '/'     Z'"' 


tp    K'"'  _    r<p 
0  '^  "^  J  d 


ix 


J', 


d 

p/'    R"~ 

<P 

d 

f'/' 

Z""  1 

dx 

L^'  •  7r_ 

e 

"dx 

L^-  /.J 

> 

'y'Tl 

-H 

'."- 

d 

'Jx 

e 


The  right-hand  member  of  this  equation  is  equal  to  a  constant,  and 
consequently  the  left-hand  member  must  also  be  a  constant.  This 
last,  however,  cannot  be  true  if  the  order*  of  the  remainder  i?'"'  is 
=  —  (;^  +  i).  If  the  {n  +  1)^*  partial  denominator  of  the  develop- 
ment of  W^in  a  continued  fraction  is  then  not  linear  (as  the  preced- 
ing ones  are),  but  of  the  degree  p  —  i,  then  7?'"'  will  be  of  degree 
—  («  -\-  p  —  i),  and  consequently 


{17) 


'^^^y^Tx 


-0{.x) 


n- 


const. 


*  By  order  of  the  transcendental  function  R(")    we    mean,   with   Heine,   the  least 
negative  exponent  in  the  development  of  i'?(«)  in  series. 


1 84 


LINEAR   DIFFERENTIAL   EQUATIONS. 


rpix) 
It  follows  from  this  that  R^"^  'iii'x  actually  represents  a  second  solu- 
tion of  the  given  differential  equation. 

The  fact  of  ^'"^  being  of  degree  —  {ri  -\-  p  —  i)  gives  sufficient 
data  for  the  determination  of  the  coefficients  of  0(;r) ;  for  in  the 
product  J/,  j^  the  terms  in  ;r" ',  ;i;"  %  .  .  ,  ,  ;ir~'"+p  "^' must  drop  out. 
Writing  now 


•  +  ?+ 


we  have  at  once  the  following  equations  of  condition 
aji„  +  aji„  _  I  J^  ,  ^  ,  J^a„j^Ji^ 


(78) 


aji„ 


=  o; 
=  o; 


^n-{-p  —  2'^«   n~  "-K-l-P  —   1        «   —    I    ~P      •      •      *       ~T"    ^2«+p  —  2  '^O    O. 

The  quantities  a  have  the  form 

a^  =  a^^^C,  +  «'='C,  +  .  .  .  +  «(P-')(rp_  ,; 

where  the  coefficients  a  are  again  linearly  expressible  in  terms  of  the 
primitive  moduli  of  periodicity  of  the  Abelian  integrals  which  enter 
into  the  problem.  From  the  first  n  of  equations  (78)  we  can  now  ex- 
press the  coefficients  h^ ,  h^ ,  .  .  .  ,  //„  as  functions  of  these  moduli, 
and  of  the  ratios 


C. 


a 


a 


c      '    r      ' 

v^p  —    I  '-'p  —    I 


c 


and  consequently  also  as  functions  of  the  ratios 


d. 


p  -  3 
^0-2 


Introducing  the  found  values  of  /^„ ,//,,,..  ,  //„  in  the  remaining 
p  —  2  equations  of  (78),  we  have  a  system  of  equations  giving 

{n  +  i){n  +  2)  ...(;/  +  p  -  2) 


I  .2.3 


(p-2) 


EQUATIONS   OF  SECOND    ORDER— RIEM ANN S  F-FUNCTION.     1 85 
'  groups  of  values  for  the  ratios 

A"  '        /^f  '        '     *     '     '        r^  ' 

Op  _  2  "p  —  2  ^p  —  2 

and,  from  equations  (72),  the  same  number  of  groups  of  values  for 
the  ratios  of  the  constants  C. 

The  results  of  the  investigation  may  be  summed  up  as  follows : 

If  the  regular  differential  equation 

has  a  rational  integral  function  of  degree  ;/  as  its  first  solution,  then 
this  function  is  the  denominator  of  the  «"'  convergent  in  the  de- 
velopment of  the  function 


'=--'  J^i  (•^--)^'(^) 
in  a  continued  fraction,  where 

(Mil)  (2.(=^)        ■  Qi(lp) 

tpi^;)  =  o,  d{z)  =  {z  -  s,)  ^  (^'U^  -  ^.)  ^'^^=^ . . .  (^  -  ^p)  ^'(^'•>' , 

and  where  the  constants  C  are  so  determined  that  the  first  Ji  partial 

denominators  of  the  continued  fraction  are  linear,  while  the(;/  -(-  i)^' 

fix) 
is  of  des^ree  p  —  i.     Further,  the  remainder  multiplied  by  -xy—:  is  a 

second  solution  of  the  same  differential  equation. 

In  particular,  if  the  quantities  7//^\  are  proper  fractions,  then 

the  second  solution,  jf^  ,  contains  no  higher  transcendents  than  entire 
Abelian  integrals  of  the  third  kind. 

It  is  not  the  intention  here  to  go  into  any  extended  account  of 
the  differential  equations  which  give  rise  to  Spherical  Harmonics 
and  the  other  allied  functions  of  Analysis,  but  it  is  interesting  to 
show  how  these  equations  are  connected  with  the  differential  equa- 
tion satisfied  by  Riemann's  P-function.    (We  will  use  this  form  of  the 


1 86 


LINEAR  DIFFERENTIAL  EQUATIONS. 


letter  P  to  denote  Riemann's  function,  in  order  to  avoid  confusion 
when  the  P  of  ordinary  zonal  harmonics  has  also  to  be  employed.) 

Riemann's  definition  of  the  P-function  is  as  follows : 

Denote  by 


b 

c 

^ 

y 

/?' 

Y 

a  function  of  x  satisfying  the  following  conditions  : 

(I)  For  all  values  of  x  other  than  a,  b,  c,  the  function  is  uniform 
and  continuous. 

(II)  Between  any  three  branches  of  the  function,  say  P',  P",  P'",, 
there  exists  a  linear  relation  with  constant  coefficients,  viz. : 

c'Y'  +  c"V"  +  c"'Y"'  =  o. 

(III)  The  function  can  be  put  in  the  form 

r.P^+G'P^'     {r  =  a,a',f3,f3',y,y') 
where  the  products 

^{x  —  a)--" ,     P^'(;tr  —  o-)- ^'   (o-  =  a,  b,  c) 

are  uniform  for  ;ir  =  o",  and  are  not  equal  either  zero  or  infinity. 

With  reference  to  the  six  real  quantities  a,  a',  ft,  ft' ,  y,  y' ,  we 
assume  that  none  of  the  differences  a  —  a\  ft  —  ft' ,  y  —  y'  are  in- 
tegers, and,  further,  that  a-\-a'-\-ft-\-ft'-\;-y-\-y'^i.  It  may 
be  left  as  an  exercise  to  the  student  to  show  from  the  point  of  view 
of  the  linear  differential  equation  satisfied  by  P  that  under  the 
above  conditions  or,  a',  ft,  ft',  y,  y'  must  be  real. 

The  first  columns  of  the  function 

'a  b  c 
a  ft  y 
a'      ft'      y' 

can  be  interchanged  among  themselves,  also  a  can  be  interchanged 
with  a  ,  ft  with  ft' ,  and  y  with  y,  y' .     Further,  let 

ex^f 


X    = 


gx^ir 


EQUATIONS  OF  SECOND    ORDER— RIEMANN S  T-FUNCTION.     1 87 

where  the  constants  are  such  that  for  x  =  a,  b,  c  vjq  have  x'  =  a',. 
b\  c' ;  then 

'  2'       b'     c' 
a       /3       y       X   \  =V   \    a       fi      y       x 


I  a'      ft'       y' 


v  a      p      y 


It  is  shown  by  Riemann  that  all  the  P-functions  having  the  same, 
values  of  a,  a',  /3,  /?',  y,  y' ,  can  be  led  back  to  the  function 

O        00        I  ] 

I 
T  \   a       ft       y       X  )-  , 

I  a'      ft'      y'  J 

which  is  denoted  more  briefly  by  the  symbol 


\o'      fi'      y'         I 


In  such  a  function  the  quantities  in  the  exponent-pairs  {a,  «■'), 
(/?,  ft'^,  {y,  y')  can  be  interchanged,  and  the  pairs  themselves  can  be 
interchanged  if  we  replace  x  by 

gx^/r 

which  for  the  first,  second,  and  third  exponent-pairs  gives  for  the 
variable  the  respective  values  O,   co,  i.     In  this  way  the  functions 


'a       ft       y 
PI  '         X 

^a        ft        y 


can  be  expressed  by  P-functions  of  the  arguments 


I        X  —  \ 


X,      I  —  X, 


X   —    I  I 


% 


•J  88  LINE  A  A'   DIFFERENTIAL   EQUATIONS. 

•and  having  the  same  exponents,  but  having  them  arranged  in  dif- 
ferent orders.     From  the  definition,  it  is  easy  to  see  that  we  have 


X 


a     b    c 
'^  \   a    ^    y      X 
.  oc'  ft'  y' 
and,  consequently,  also 


'a    (3    y 
x\\-xyV{  X 

a'  ft'  y' 


X  —  b. 


1 


=  Y  \    a  ^d    ft    -^$    y    X    \ 
la'^S    ft'-^d    y'        J 


_  p  #  '"^  +  ^'  /^   -6  -  e,y  -f  e  ^ 

~         //  ^S,  ft'  -  d  -  e,y'  -^e 


Through  these  transformations  one  sees  that  the  two  exponents 
in  the  different  pairs  can  take  arbitrary  values,  the  sum 

«  +  «'+...+;/' 

remaining,  however,  unity.      For  any  set  of  exponents  we  can  then 
replace  any  other  set  provided  the  differences 

a—  a',     ft  —  ft',     y  —  y' 


remain  the  same.     All  functions  of  the  form 
x^{\  —  xW 


p     y 
ft'     y' 


are  represented  by  Riemann  by  the  symbol 

T{a-a',     ft -ft',     y-y',     x). 

From  what  precedes  the  differential  equation  satisfied  by  P  is 
at  once  seen  to  be  a  regular  linear  differential  equation  of  the  second 
order,  having  a,  b,  c  as  critical  points,  and  whose  integrals  contain 
no  logarithms.     The  equation  is  then  of  the  form 


(79)     Tx^- 


Ax'+Bx-{-C        dy    ,       Fx'^-Gx^H 

-JZ  +  /^    ^w-^    MV-.    .\-^y  =  o- 


{x—d){x—b){x—c)  dx    '    {x—df{x~by{x—cy 


(As  an  additional  exercise  the  student  may  show  that  the  numer- 
ator of  the  coefficient  of  y  cannot  be  of  a  degree  higher  than  the 


EQUATIONS  OF  SECOND    ORDER— RIEM ANN' S  T-FUNCTION.     1 89- 


% 


second.)  The  indicial  equations  corresponding  to  the  points  a,  b,  c 
are  as  follows : 

Aa^^Ba-\-C   ,    Fa^^Ga-\-H 

^  ,      Ab'+Bb^C    ,    Fb'+Gb+H 
(6)     For  X  =  b,      rir—  i)+r  -77 ^7 r  +  u: Wl ^^  =  o ; 

(v)     For  X  =  c,     r(r—i)-\-r rj- — rr-  +  -7 ^i7 r-r^  =  O. 

From  (III)  it  is  obvious  that  the  roots  of  {a)  are  {a,  a'),  the  roots  of 
(/?)  are  (y^,  /?'),  and  the  roots  of  {y)  are  (7,  ;^').  We  have  then  for 
these  roots  the  following  equations  : 


(80) 


I 


(81) 


or 


(82) 


,       {a  -  b){a  -  c)  -  Aa'  -  Ba  -  C 
'  {a  —  b)\a  —  c) 

,  _{b  -  a)ib  -  c)  -  Ab'  -  Bb  -  C 
^^^^  -  {b  -  a){b  -  c)  ' 

^       (^  -  a){c  -  b)  -  Ac'  -  Be  -  C ^ 


r  +  r 


(c  —  a){c  —  b) 


13ft'         = 


Fa"  -^  Ga  +  H 

{a  —  by  {a  —  cy 

Fb'  -^  Gb  -\-  H 
{p  -  a)\b  -  cf ' 

Fc'  -{-  Gc-\-  H 


jy         -   ic-a)\c-by' 

{Aa'-{-  Ba  +  C){b  -  c) 


or  +  or'  =  I  + 


{a  -  b){b  -  c){c  -  a)    ' 

{Ab'  +  Bb  +  C){c  -  a) 
{a  -  b){b  -  c){c  -  a)   ' 

,_  ^,{A^  +  Bc  +  C){a-by 
J+r  -  I  +  7^^3)(^^r7)(^  _  a)    ' 


390 


LINEAR  DIFFERENTIAL   EQUATIONS. 


(83) 


Write 


\fi^'        = 


rr       = 


(Fa'  ^  Ga  -\-  I/){d  -  c)' 
[a  -  b)\b  -  c)\c  -  ay     ' 

{Fb'  +  g^  +  H\c  -  ay 

{a  -  by{b  -  cy{c  -  ay  ' 

{Fc'  +  Gc  ^  H){a  -  by 
{a  -  by{b  -  cy{c  -  a)^  ' 


^  =  «+«'  +  /?  +  /?'  +  r  +  /; 

then  on  adding  equations  (82)  and  observing  that 

a\b-c)  +  b\c-a)  +  c\a-b)  =:  -  {a-b){b-c){c-a), 

a{b—c)  -{-   b{c—a)  -{-   c{a  —  b)  =  o, 
{b—c)  -\-     ic—a)  -\-     (a—b)  =  o, 
we  have 
(84)     A  =  3  -  2,     =i-a-a'-}-i-/3-  fi'-\-  i-y  -  y' . 

Multiplying  the  first   of  equations  (82)  by  a,  the   second  by  b,  the 
third  by  c,  adding,  and  observing  that 


\b  -  ^)  +  b\c  -  rt)  +  c\a  -  b) 


{a+b^c), 


{a  —  b){b  —  c){c  —  a) 
we  have 

(85)  B=-l{i-a-a')(b-^c)^{i-^-^'){c+a)-^{i-y-y'){a+b)\. 
In  like  manner  we  find  for  C  the  form 

(86)  C  ={\-cx-  a')bc  +  (I  -  yS  -  fi')ca  ^  {\  -  y  -  y')ab. 
Multiply  now  equations  (83)  in  order  by 

(c-a){a-b\     {a-b){b-c\     {b  -  c){c  -  a\ 
and  adding,  we  get  the  value  of  F\  multiplying  again  by 

{c-a){a-b){b^c\     {a-b){b-c\c-^a\     {b-c){c-d){a-^b\ 


I 


t 


I 


EQUATION'S   OF  SECOND   ORDER— RIEM ANN' S  F-EUNCTION.      I9I 

and  adding,  we  get  G ;  finally,  multiplying  by 

{c  —  a){a  —  b)bc,     {a  —  b){b  —c  )ca,     (b  —  c){c  —  d)ab, 

and  adding,  we  get  H.     The  values  of  F,  G,  H  thus  found  are 

F  =  —  \aa\c  -  d){a  -  ^)  +  /?/i'(«  -  i>){b  -  c) 

-\-yy\b  -  c){c  -a)\, 

G  =  { aa\c  -  a){a  -  b){b  +  ^)  +  /?/?'(«  -  b\b  -  c){c  +  a) 

^yy\b-c){c-d){a^b)\, 

H  =  -\  aa'{c  -  a){a  -  b)bc  +  §^\a  -  b){b  -c  )ca 

H~  yy'ip  ~~  ^){^  ~  ^)<^b\' 


m    \ 


The  values  oi  A,  B,  .  .  .  ,  H  from  (84),  (85),  (86),  and  (87),  substi- 
tuted in  equation  (79),  give  us  for  this  equation  the  form 


d-y 


f  (l  -  a  -  a')\_x'  -ib  -\-  c)x  +  be] 

+  (i  -  /5  -  /J')[^^  -  (^  +  ^k  +  ^«]  1  dy 


J 


I 


+ 


{x  —  d){x  —  b){x  —  c) 

aa\a  -  b){a  -  c){x'-  {b  +  c)x  +  ^^] 

+  ftli'ib  —  c){b  -  a)[x'  -  (r  +  a)x  +  r^] 

+  yy{^  —  ^)(^  —  ^)[-^'  —  (^^  +  ^)-^ + ^^] 
{x  —a  )-{x  —  b)\x  —  cj 


y  =  o\ 


or 


+ 


(;ir— «)(;ir— <^)(;f— <r)  (  ;ir— « 


aa'{a-b){a-c)    ,    fift'{b-c){b-d) 


;f — c  ) 


This  form  of  the  differential  equation  satisfied  by  the  P-function 
was  first  given  by  E.  Papperitz  ;  *  Riemann  has  given  indications  as 

*  Ueber   verwandte  s-Functionen,  von  Erwin  Papperitz.     Math.  Annalen,  vol.  xxv. 
p.  212. 


192  LINEAR  DIFFERENTIAL  EQUATIONS. 

to  the  formation  of  the  equation,  but  does  not  seem  to  have  worked 
out  the  form. 

In  (79)  make  ;jr  =  -  ;  the  equation  then  takes  the  form 


^^^       dt-^Xt       t{\  -  at){i  -  d^){i  -  ct)  \  dt 

+  (I  -  at)Xi  -  bi)\\-cty^~  °' 

The   coefificient  of  y   does  not  contain  /  in  the  denominator  as  a 

dy 
factor;  the  coefficient  of  -r-  is 

at 

I      2(1  —  at){\  —  bt\\  —  ct)  —  A  —  Bt  —  Cf  ^ 
1  '  (i  —  at){\  —  bt){i  —ct)  ' 

substituting  for  ^4,  B,  C  their  above-found  values,  we  have  for  the 
numerator  of  this  coefficient  the  value 

2(1  -at){i-bt){i  -ct)-{z-  a-a'-/3-/3'-r-r') 

+  [(i-a-a'){b+c)+(i~/3-^'){c-^a)^{l-r-r'){a^b)]t 

-[{1  -a-  a')bc  +  (I  -  /i  -  f3')ca  ^{i-y-  y')aby. 

Now  remembering  that  a  -\-  a'  A^  ft  A^  fi'  ^  y  -^  y'  =  \,  v.^^  see  that 
this  last  expression    is  divisible  by  /,  and    consequently    that    the 

dy 
coefficient  of  -~-  in  (90)  does  not  contain  /  as  a  factor  in  its  denom- 
inator. It  follows  then  that  /  =  o  is  not  a  critical  point  of  (90),  and 
consequently  that  ;t-  =  00  is  not  a  critical  point  of  (88)  (or  (89)).  The 
points  a,  b,  c  are  then  the  only  critical  points  of  our  differential 
equation. 

If  in  (88)  we  write 

a=—  \,     b=~,     ^  =  -1-1, 

a  =  a'  =  y  =  y'  =  o,      ft  =  —  n,      ft'  =  ?t-\-  i,     lim  e  =  O, 


I 


EQUATIONS  OF  SECOND   ORDER— RIEMANN' S  P-FUNCTION.      1 93 

we  find,  after  simple  reductions,  the  equation  for  zonal  spherical 
harmonics,  viz., 

d^'P  2x      dP        n{n-\-\) 

(90  ~d^  -  T^^^-d-x^  'V^l^^-''' 

Again,  writing 

e 
a  ^=  V,     a'  =  —  V,     yS  =  tGO,     ^'  =  —  ico, 

Hm  e  =  o,     lim  03=00, 
we  have  the  differential  equation  for  Bessel's  functions,  viz., 

r      \  d^J    ,     \     dj    ,     {  v^\    r 

(92)  — ^-^ r-+ii  —  —  /  =  0- 

^^  ^  dx^    '    X  dx    ^     \  x'J 

If,  in  space  of  three  dimensions,  we  consider  two  points,  one  at  a 
distance  unity  and  the  other  at  a  distance  p  from  the  origin,  and  if 
GO  denote  the  angle  between  the  lines  drawn  from  these  points  to 
the  origin,  we  have  for  the  reciprocal  of  the  distance  between  the 

points 

(93)  i  =  (i  -  2/)  cos  a?  4-  p')-K 

The  ordinary  zonal  harmonics  are  the  coefificients  of  the  different 
powers  of  p  in  the  development  of  (93)  in  ascending  powers  of  p, 
viz., 

(94)  \    =    Po+  P.P  +  P.P'  +   "     '    +  PnP"   + 

Writing  cos  go  =z  x,  these  functions  P„  satisfy  the  differential  equa- 
tion 

/   -.\  ^y  2;i-       dy     ,    nin  +1) 

ax         \  —  x^  dx  I  —  X' 


194  LINEAR  DIFFERENTIAL   EQUATIONS. 

The  second  integral  of  this  equation  is  the  spherical  harmonic  of  the 
second  kind,  P„  being  the  zonal  harmonic  of  the  first  kind. 
From  Laplace's  definite  integral  form  for  /*„ ,  viz., 

P„=  ^  J    {x  -\-  Vx-—  I  cos  <p)d(p, 

we  derive  that  P„  is  the  term  independent  of  0  in  the  development 
of  {x  -|-  Vx''  —  I  cos  (py  in  a  series  going  according  to  the  cosines 
of  multiples  of  0.  The  coefificient  of  cos  w0,  viz.,  P„,  ,„,  is  the 
associated  /miction  of  the  first  kind  of  degree  n  and  order  m.  This 
function,  as  is  well  known,  satisfies  the  differential  equation 

,    „  d'^y  2x      dy     ,    n{n-\-\){\  —  x^)  —  vi" 

(q6)  —^  — 4~  +  ^ — --^^ '.-^— —  y  =  o. 

^^  '  dx'        I  —  X'  dx    '  (I  —xj  -^ 

The  second  integral  of  this  equation  is  the  associated  function  of  the 
second  kind,  and  is  denoted  by  ^„,  ,„.  If  instead  of  n  being  an  inte- 
ger it  is  of  the  form 

n  =  p.  —  ^, 
then 

n-\-\  =  /I  +  f , 

and  (97)  becomes 

dy  2x     dy       (/.--i)(i-,t-)-;;.- 

which  is,  Basset's  Hydrodynamics,  Vol.  II.  page  22,  the  differential 
equation  for  Hicks's  Toroidal  Functions.  The  spherical  harmonics 
for  the  cases  above  considered  present  themselves  in  the  theory  of 
attraction  in  space  of  three  dimensions  ;  we  will  speak  of  them  as 
being  of  rank  2.  In  considering  the  theory  of  attraction  in  space  of 
k  -\-  \  dimensions,  we  get  spherical  harmonics  of  the  rank  k\  these 
harmonics  appear  as  the  coefficients  of  the  powers  of  0  in  the  de- 
velopment of 

(I  -  2xp^r  P")      '  • 


EQUATIONS  OF  SECOND    ORDER— TOROIDAL   FUNCTIONS.      1 95 
The  differential  equation  of  these  functions  is  known  to  be 

d'^y  kx    dy        n{n -\- k  —  i){l  —  x")  —  ■m{m -\- k  —  2) 

For  k  =  2,  let  X  =  cos  — ,  and  then  for  an  indefinitely  increasing 
value  of  n  we  get  the  Bessel's  functions, 


r  2«  /       M 

A  =  hm  --=P„^„  \cos  -J , 

100)  \ 


.                       2-"-'4/;r  /        ^, 

I    F,„  =  hm — — •  Q„^  ,„  I  cos--]  ; 

the  differential  equation  for  which  is 

dy       I  dy        (         7n'\ 

<^°^)  d¥  +  ^d^+V-wl^  =  ''' 

The  corresponding  equation  for  the  Bessel's  functions  of  rank  k  is 
dy    ,    k—\dy    ,    /         mim -\- k  —  2)\ 

<I02)  ^^-+-^-^+1^-  ^« j^  =  0. 

Equation  (98),  the  general  equation  for  toroidal  functions,  is  the 
only  one  of  the  preceding  equations  involving  a  parameter  which  is 
other  than  an  integer.  In  (98)  replace  //  by  «,  and  make  ;;z  =  o ; 
we  have  now  the  differential  equation  of  zonal  toroidal  functions, 
viz., 

d'^y  2x     dy    ,    «*  —  4- 

Hicks  in  his  memoir*  gives  this  equation  in  the  form 
<I04)  ^^  +  coth  u^£-{re-^)y=o; 


Phil.  Trans,  of  the  Royal  Society;  Part  III.,  1881,  p.  617,  equation  (9). 


196  LINEAR  DIFFERENTIAL   EQUATIONS. 

in  this  writing 


X 


X  =  cosh  u,      Vx"  —  I  =  sinh  u,     —  jz  =  coth  //, 

Vx'-  I 

we  arrive  at  (103).     It  is  evident  that  the  Jirst  integral,  say  P„,  of 

2«  -4-  I 
(104)  is  a  zonal  spherical  harmonic  of  degree with  a  pure 

imaginary  for  argument.     Writing,  for  brevity, 
C  =  cosh  ti,     S  =  sinh  u, 
Hicks  finds  for  the  integral  P„  the  form 

and  shows  that  the  three  integrals  P„ .  i,  P„,  P„  +  i  are  connected 
by  the  sequence  relation 

(106)  (2«  +  i)P„  +  ,  -  4^iCP„  +  {2n  -  i)P„  _ ,  =  o. 

By  aid  of  this  equation  Hicks  finds  the  value  of  P„  as  follows  :  In 
(106)  write 

_      (2;/  -  2){2U  -  4)    ...    2 

^"-(2;.-i)(2«-3)  ...  3.1    "' 

with  P,  =  ii, ,     P^  =  i{^ ; 

(2;/  +  iV 
then  «„  +  2  —  20/'„  +  i  H 7 j — r  ?/„  =  o ; 

or,  writing 


l)   —  I  '2  \2«         2n  4-  2/ 


"  ~~  2n{27i  +  2)  '     ""  (2«  4-  0'  "~  I  '     ~  2  \2«  2«  -f 

and  ^0=2, 


w 


'EQUATIONS  OF  SECOND   ORDER— TOROIDAL   FUNCTIONS.      1 9/ 
It  is  clear  from  this  that  «„  is  of  the  form 

<^mU,  —  fi„u^ ; 

where  a„ ,  ^„  are  rational  integral  functions,  algebraic  functions,  of 
2C;  a„  o(  degree  n  —  i,  and  y5„  of  degree  n  —  2.  Writing  2C  =  t, 
nve  have 


(107) 


u^  =  u,; 

u,  —  u^ ; 

u^  =  tu^  —  \u^ ; 

?^3  =  (^'  —  c^Ux  —  ¥k  • 


We  can  now  show  that  a„ ,  /?„  are  of  the  form 

■(108)     a^  =  a„J''-^  +  a„J"-^  +  a„J"-^+  .  .  .  +  ^„^/«-^'--' +  .  .  . 

For,  supposing  a„  of  this  form,  i.e.,  wanting  every  other  power  of  t, 
it  follows  at  once  that  a„j_^  is  of  the  same  form;  it  is  seen  above 
that  0-3  is  of  this  form,  and  so  the  statement  is  generally  true.     The 
same  remarks  apply  to  the  function  y5„ . 
Now  a„  satisfies  the  equation 


<I09) 


^(^n  -  I  ^K  -  2  ^n 


with  a'„  =  o,  a-,  =■  I.     Hence,  substituting  the  above  value  for  a„, 
we  must  have 


Cl^^    t/« 


—  c„_^a 


n  -  2"'n  -  2,  r  -  I  » 


^no    ^«  -  1,  o  


t»  (no) 
also, 
(III) 
Hence 

(112)        a„^=    —   (^»  -  2«,«  -  2,  r  -  I  +  '^^  -  3««  -  3,  r  -  I  +   •    •   •   +  ^2r  -  I  «2r  -  I,  r  -  1). 

From  this  we  have 

««i  =  -  (^«  -  2  +  ^„  -  3  +  •  •  •  +  ^:) ; 

««2  =  ^«  -  2(^«  -  4  +   •    •    •    +  ^i)  +  ^n  -A^n  -  5  +   .    .    .    +  ^.)  +    .   .    .    +  C,C,  . 


I 


198  LINEAR  DIFFERENTIAL  EQUATIONS. 

From  this  last  we  observe  that  a„^  is  equal  to  the  sum  of  the 
products  cfj  taken  two  and  two,  with  the  exception  of  all  products 
where  the  subscripts  are  successive. 

Assume  that  (— )''«„^  =  sum  of  products  of  the  ^'s  taken  r  at  a 
time  up  to  ^„_2,  excepting  such  products  in  which  successive  sub- 
scripts occur.     Then 

«„,r+i  =  (— )''^'l^«-2(prod.  up  to  ^„_4,  r  at  time,  etc.,     ...     ) 

+  ^„-3(        "  "  ^»-5  "  "  •     •     •        ) 

+     .     ..  ...         I 

=  (—)'■+' I  prod,  {r -\-  i)  at  a  time  up  to  ^„_2  without  succes-^ 
sive  subscripts.  [ 

Whence  by  induction  the  assumption  is  seen  to  be  universally  true 
It  may  be  thus  stated  :  «„^  is  the  sum  r  at  a  time  of  the  terms 

3^         5^        r  i?n-iy 


2.4'     4.6'     6.8'    '  '  '   '   {2n-  2){2n  -  4)' 

all  products  being  thrown  out  in  which,  regarding  the  numbers   in 
the  denominators  as  undecomposable,  a  square  occurs   in  the  de- 
nominator. 
We  have 

««o  =  I  ; 

(4«  -  3)(«  -  2) 


««i  =  — 


4(«  -  i) 


This  result  is  of  very  little  use  for  application.  If  the  coefficients 
a„  are  needed  for  particular  values  of  t,  they  can  be  very  rapidly 
calculated  by  means  of  equation  (109),  while  if  their  general  values 
are  to  be  tabulated,  equation  (no)  will  serve  to  calculate  them  in 
succession. 

Further,  2/?„  is  the  same  kind  of  function  as  a^  in  every  way, 
except  that  it  does  not  contain  c^ ;  in  fact,  2y5„  is  the  same  function 
of  <;, ,  /Tj ,  .  .  .  ,  ^„  .  2  that  «'„  _ ,  is  of  <;, ,  r, ,  .  .  .  ,  ^„  _  3 .  Calling  this 
«'«-!>  we  can  then  write 


EQUATIONS  OF  SECOND   ORDER— TOROIDAL   FUNCTIONS.      I99 

The  functions  ;/„  and  ;/,  are  expressible  as  elliptic  integrals,  viz.: 
Writing 

25 


or 


and 


-  -CT5'   ^''-{c^sy 


,     (.-.  k^  +  k-=i,) 


t  =  2C  =  k'  -\-  y  , 


I 


we  have 


(113) 


dd 


VC  —  5  COS 


=  iV'k' 


=    2\/k'F\ 


do 


Vi-  k'  sin^ 


\/C-  Scose.dd 


V^Vo 


Vi  -  k'sin'  e.dd 


Hence 


(114)  P„=2 


{2n  —  2){2n  —  4)  ...  2 


{2n  -  i)(2«  -  3)  ...   i[_\/k 


^  77  _  1  i/b'    a'       F 


For  n  =.  o,  P „  =^  7t  \  for  n  =  00,  P„  =  co. 

By  very  simple  operations  the  second  integral  Q,^  can  be  shown 
to  have  the  form 


(115)  Qn  =  2 


{2n  —  2)  ...  2 
(2;/  —  i)  ...  3 


LVk 


^^(^F'  -E')-\a\_Jk'F' 


F'  and  E'  being  the  complete  elliptic  integrals  of  the  first  and  second 
kinds  with  modulus  k' .  The  functions  P„  and  Q^  thus  found  can 
obviously  be  put  in  the  forms  of  the  hypergeometric  series.  For 
further  information  on  the  Toroidal  Functions  the  reader  is  referred 
to  Hicks's  memoir,  and  also  to  Basset's  Hydrodynamics,  Vol.  H. 


200  LINEAR  DIFFERENTIAL   EQUATIONS. 

Chap.  XII.  It  is  interesting  to  derive  the  above  results  directly 
from  equation  (103),  but  limits  of  space  will  not  allow  us  to  enter 
into  the  investigation. 

Returning  now  to  equation  (99),  we  will  write  it  in  the  form 

d^y  fx     dy     ,   ^(^-f-/-i)(i-^-)-/,(/,-f/-2) 

(^^^)    -dx^-T—eTx-^- xr=r^j -^'  =  °' 

where  the  parameters  /",  g,  h  are  no  longer  restricted  to  integral 
values.*  We  will  define  now  as  spherical  harmonics  of  the  first  and 
second  kind  of  rank  /,  degree  g,  and  order  //,  two  definite  linearly- 
independent  particular  integrals  of  (i  16). 

These  generalized  functions  therefore  depend  upon  the  three 
arbitrary  parameters  /,  g,  h,  and  have  the  points  -\-  i,  —  i,  00,  as 
critical  points. 

The  P-function  above  defined  depends  essentially  on  the  differ- 
ences of  the  pairs  of  exponents  {a,  a'),  {fi,  /?'),  (;/,  y'),  and  upon  the 
critical  points  a,  b,  c.  At  first  sight  it  might  appear  that  these  new 
functions,  i.e.,  the  integrals  of  (116),  were  identical  with  the  general 
P-functions.  That  this  is  not  the  case  can,  however,  be  easily  seen. 
The  differential  equation  of  the  P-function  is,  equation  (89), 

i...\     ^>    ,    Si- PC- a'   ,    I- ft-  ft'    ,    i-y-Y'\dy 
^  ^^     dx''^\       x-a      "^       x-b       "^       x-c       )  dx 

I  (  a  a'  {a  —  b){a  —  c) 


'    {x  —  a){x  —  b){x  —  c)  \  X  —  a) 

,     ^^'{b-c\b-a)     ,     yy'{c-d){c-b)^ 


This  is  changed  into  (116)  by  the  following  substitutions: 

'«=  —  I,  b  ■=  CO,  c  ^  -\-  I  ; 

h  h 

(118)  j "  =   2-'  ^  =  --^'       y  -^    2' 

*  Studien  iiber  die  Kugel-  und  Cylinder functionen,  von  R.  Olbricht.     Halle,  1887. 


EQUATIONS  OF  SECOND  ORDER— GENERALIZED  HARMONICS.    201 
We  have  now  the  P-function 


<ii9)    P 


-  I, 

2' 


00, 


+  1 


^-^,  ^+/-.  -^^,^ 


In  this  the  differences  of  the  pairs  of  exponents  are 


I 


h-V 


/-2 


2^-/+  I,     h  + 


/- 


two  of  which  are  equal.     We  arrive  thus  at  the  important  theorem  : 

»  The  theory  of  the  generalized  spherical  harmonic  is  identical  with 

the  theory  of  the  ^-function,  in  which  the  differences  of  two  pairs  of 
■corresponding  exponents  are  equal. 

We  will  merely  state,  without  proving,  two  other  theorems  found 
by  Olbricht,  referring  the  reader  to  his  memoir  for  the  proofs  and 
also  for  other  interesting  results  concerning  the  generalized  spherical 
harmonics  and  Bessel's  functions. 

ia)  The  generalised  spherical  harmonic  of  rank  f,  degree  g,  and 
crder  h  are  derived  from  the  'P-function 


P\ 


—  I 

00 

+  1 

h' 

2 

g' 

h' 

2 

h' 
2 

<r'+l' 

_  h' 

2 

■which  represents  the  spherical  harmonics  of  rank  z,  degree  g' ,  and 
order  h'  by  replacing  g' ,  h'  by 


s'  =  s  +  ^-^,   k'  =  h+-L-^ 


202  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  multiplying  the  result  by 

{B)  The  generalised  Bessel  's  functions  are  obtained  from  those  of 
the  second  order  by  multiplying  the  'P-function 


lim  P 


gn 


ig 


00  O 

ig  h 

—  ig     —  h 


2-/ 


by  X   ""    ,  and  replacing  h  by 


i|/(2-/)^ +/</.+/ -2). 

In  the  next  chapter,  M.  Goursat's  thesis,  will  be  found  an  investi- 
gation of  the  differential  equation  of  the  second  order  satisfied  by 
the  complete  elliptic  integral  K. 


At  the  time  of  writing  the  preceding  pages  the  author  had  not 
seen  an  interesting  paper  by  Humbert,*  of  which  the  following  is  an 
account,  and  did  not  know  that  some  of  Heun's  results  (see  above) 
had  been  previously  given.  Humbert  investigates  the  most  general 
polynomials  satisfying  the  equation 


(I20) 


^•("^'^  %  +  ^'(^)^  +  Q^^'^^y  =  ° 


where,  as  before,  f,  <2, »  Q2  are  polynomials  of  the  degrees  p,  p  —  i, 
p  —  2  respectively. 

If  we  wish  (120)  to  have  as  a  solution  a  polynomial  in  x  of  degree 
n,  the  three  functions  ip,  (2, ,  Qi  cannot  be  arbitrarily  taken.     Heine 

*  "  Sur  I'equation   differentielle  lin^aire  du  second  ordre,"  par  M.  G.   Humbert. 
Journal  de  VEcole  Polytechnique,  cahier  48,  1880. 


EQUATIONS  OF  SECOND   ORDER— POLYNOMIAL  INTEGRALS.    203, 

has  shown*  (see  above)  that  if  ^(;tr)  and  Q^x)  are  arbitrary  poly- 
nomials of  degrees  p  and  p  —  i  respectively,  there  -exist  only 

(n-\-\){n-\-2)  .  .  .  {n-\-p—2) 
I  .  2  ...  (p  -  2) 

integral  polynomials  Q^x)  of  degree  p  —  2,  such  that  equation  (120) 
admits  as  a  solution  a  polynomial  in  x  of  degree  n. 

Denoting  hy  x^,  x^,  .  .  .  ,  Xp  the  finite  singular  points  of  the 
differential  equation,  we  write,  as  before, 

^•(^)  =  {x  —  x,){x  -  X,)  .  .  .  {x  -  Xp); 

also  write 


Qix)  =  iix) 


M. 


+  j^  +  ---  +  j^\ 


a  polynomial  of  degree  p  —  i.     Defining  a  new  function  K{x)  by 
the  equation 

we  have 

K{x)  ={x  —  xy^  (x  —  x^^   .  ,  .  (x—  Xpfp . 

This  function  IC{x)  has  an  important  property;  viz.,  if  we  make  in 
(120)  the  substitution 

I 

this  equation  retains  the  same  form  ;  it  becomes  in  fact 

(123)    i<x)  ^  +  [2tp\x)  -  Q,{x)-]  '^^-^lQlx)-rp"{x)-Q:{x)]u=o.. 


*  Handbuch  der  Kugelfunctionen,  zweite  Auflage,  p.  473. 


204  LINEAR  DIFFERENTIAL  EQUATIONS. 

For  simplicity  we  will  first  assume  p  ^  3 ;  we  have  then 

tl.\x)  =  {x  -  x^{x  -  x^{x  -  x^ ; 
QiA    __  ^'        J^        /^»  I     _./^3 


^'(a-)  ;ir  —  x^    '     Jtr  —  x^    '     ;i;  —  ;r. 

We  will  also  first  assume  that  the  constants  jx^,  fx^,  fx^  are  all  posi- 
tive. Consider  a  path  of  integration  between  the  points  ;r,  and  x^, 
and  another  going  from  x^  to  x^ ;  the  integrals 


will  then  have  a  sense  and  be  perfectly  determinate  so  long  as  these 
paths  of  integration  do  not  pass  through  the  point  x.  Suppose  now 
P„{x)  to  be  a  polynomial  of  degree  71  satisfying  equation  (120);  then 
equation  (123)  will  admit  as  a  solution  the  function 

J{x)  ^"^'^^' 

Substituting  in  the  left-hand  member  of  this  last  equation  the  func- 
tion 


we  readily  find,  by  aid  of  a  known  formula  due  to  Heine, 
(125)  iix)"^)  +  [2^'{x)  -  Qix)\  ^ 


I 


EQUATIONS  OF  SECOND    ORDER— POLYNOMIAL   INTEGRALS.    20$ 
where 


.=-.^..,,)-[i(4=^]- 


-S{x)-S{z)' 

L        X  —  Z 

I    T{x)-T{z) 


_  K{z)Plz)    ,K{z)        .  t\^)  -  S{z) 


X  —  z 


X  —  z 


'K{z) 
\-ipiz) 


Pn{z)\. 


Developing  ^,  we  find 


I 


^  ^  K{z)Plz)  _  K{z)P„\z) 
(x  —  zf  X  —  z 


This  is  to  be  taken  between  the  lin:iits  x^  and  x^ ;  and  since  K{z) 
vanishes  at  these  limits,  it  follows  that  (p  disappears  from  (115).  As 
to 


I 


fy{z,x)dz, 


we  see  that  it  is  a  constant  in  x.     We  conclude  then  that  v^  is  a 
solution  of  the  equation 

(126)         rp{x)^^-^r[2f'{x)-Qix)-]'^'' 


dx" 


dx 


+[aW+  ^"W  -  Q\^)V  =  const., 
the  constant  being  determinate.     So  also  the  function 


(127) 


v„  = 


K_(zl   P^iJl 
tp(z)    X  —  z 


dz 


I 

206  LINEAR  DIFFERENTIAL   EQUATIONS. 

is  a  solution  of  this  same  equation  with  a  different  value  of  the  con- 
stant in  the  right-hand  member. 

It  follows  now  that  v^  and  v^  are  solutions  of  the  equation 

(128)         iix)  ^-  +  \inx)  -  Qix)\  ^^ 

+  {ii'\x)  -  2q:{x)  -f  aw]  ~ 

+  [a'(-r)  +  f"{x)  -  Q:\x)]  u  =  o, 
obtained  by  differentiating  (123).     The  function  ,   /'«(•*')  being  a 

'l\X) 

solution  of  (123),  is  therefore  also  a  solution  of  (128). 

Between  these  three  solutions  of  this  last  equation  we  can  show 
now  that  there  exists  no  linear  relation  with  constant  coefificients. 

The    functions   v^    and    v^   developed    according    to    descending 

powers  of  x  commence,  at  the  highest,  by  a  term  in  — ;  on  the  other 
hand,  the  function  -77W-  P»{-^)  is  of  the  degree  i 

/^i  +  /".  +  y"3  +  «  —  3»  j 

which  is  greater  than  —  i,  since  n>  i,  and  yw, ,  /^, ,  /I3  are  all  posi- 
tive. If  then  there  exists  any  linear  relation,  it  must  be  between 
the  functions  v^  and  v^  alone,  and  so  be  of  the  form 

where  A.  is  a  constant,  or 


tfiz)    x  -  z  J       i-{z)    X-  z 


I 


EQUATIONS  OF  SECOND   ORDER— POLYNOMIAL   INTEGRALS.     20/ 
We  can  write 


m  PM  ,^  ^  ^^(,)  /  ^(^)  ^^ 


,   ^{s)  X  -z  "^  v..   'fi^)    x-z 


The  second  term  of  the  second  member  of  this  last  equation  is  an 
integral  polynomial  in  x  of  degree  ;<!  —  i  at  most.     We  have  then 

„,  ,(     r^Kiz)     dz         ,    n^i^)     dz     ] 


J7„  _  i(;r)  being  a  polynomial  in  x  of  degree  ;?  —  i  at  most.  It  is 
easy  to  see  that  the  expression  in  |  |  satisfies  the  differential  equa- 
tion 

( 1 29)  ^'(,r)  ^  =  \Qix)  -  i\x)\  y^ax+  (3, 

where  a  and  /?  are  constants.  This  equation  admits  then  as  a  solu- 
tion the  rational  function 

n„  -  ,{x) 

Pn{x)        ' 

which  after  suppressing  common  factors  will  be  written 

P{x)' 
We  shall  have  then 

an'P  -P'n)  =  (P.  -  t')Pn-{-  {ax  +  0)P\ 

and  consequently  rp  .  P'.  II  is  divisible  by  P.  Now  P{x)  is  prime 
to  II (x),  and  therefore  it  must  admit  at  least  one  of  the  factors  of  ip, 
say  the  factor  x  —  x^ .     Again,  the  factors  of  P{x)  are  also  factors 


208  LINEAR  DIFFERENTIAL  EQUATIONS. 

of  Pn{x) ;  Pn{,x)  therefore  vanishes  for  x  ^=  x^.  The  differential 
equation 

i,p:\x)  +  q,p:{x) + Q^pix)  =  o 

shows  that  P„'{^o)  =  O-  Taking  the  successive  derivatives  of  the 
first  member  of  this  equation,  we  find  in  Hke  manner  that 

p„"{x„)  =  o,  p:'\x:)  =  o,  . . . ,  p-{x,)  =  0,  . . . , 

provided  only  that  none  of  the  polynomials 

vanish  for  x  =  x^.     In  order  that  they  should  vanish  we  must  have 

which  is  impossible,  since  //,  >  o.  We  arrive  thus  finally  at  the  con- 
clusion that  the  constant  P„"{x^)  is  zero,  and  consequently,  as  is 
easily  seen,  that  the  relation 

'v^  =  Az/, 
is  impossible. 

Denote  by  Q  the  coefficient  of  the  first  term  in  Q^{^) ;  then,  as  is 
easily  seen,  the  indicial  equation  of  (120)  for  the  region  -ir  =  00  is 

(i 30)  r{r  -  I)  +  (a',  +  M.  +  M.y  +  Q  =  o. 

Since  P„{x)  is  to  be  a  solution  of  (120),  n  must  be  a  root  of  (130); 
the  two  roots  are  then 

n, 

—  Ml  —  M.  ~  M,  —  ^^  +  ^^ 

For  (128)  the  roots  of  the  corresponding  indicial  equation  are  readily 
found  to  be 

-  I,  I 

^  +  M.  +  M.  +  Mi  —  3y 
—  n  —  2. 


EQUATIONS  OF  SECOND    ORDER— POLYNOMIAL   INTEGRALS.    20g 

The  root  ?2  +  yWi  +  /<a  +  /^s  —  3  is  obviously  greater  than  —  n  —  2. 
Equation  (128)  has  therefore  a  solution  j„+2  developable  in  a  series, 
of  the  form 


I 

(131)  y„^^  =  -^- 


consequently 


(132)  y„^,  =  V,  +  co,v,  +  d  -^P„{x). 

In    this   it   is  clear   that  6  =  0,  since    the  degree  of  —j—^-PJx)  is 
greater  than  —  i.     There  remains,  then, 

i 

(133)  ^.  +   'i'l^a  =   (^-fl)  ' 

denoting  by  [— ^  j  a  series  going  according  to  descending  powers  of 


I 

x'f 
Now, 


X,  and  commencing  with  a  term  in  —^ 


+  rK{z)PM-Pn{x)^^ 

The  second  term  of  the  third  member  of  this  equation  is  an  integral 
polynomial  of  degree  n  —  i  at  most. 
Write 

f  K{z)      dz 


2IO  LINEAR   DIFFERENTIAL   EQUATIONS. 


the  relation  (133)  now  becomes 

(135)  P.  W[A  +  ooJ:\  =  i7„  . .  (;r)  +  (^) 


,  I 

order  -^^  pres,  by  the  quotient 


From  this  we  see  that  /,  +  00 J^  is  represented,  to  terms   of   the 

:nt 

^»-:(-y) 
Pnix)    ' 

In  other  words,  having  given  the  functions  /j  and  /, ,  if  we  form  the 
product 

P(;r)[A  +  ooj^l 

where  P{x)  is  a  polynomial  of  degree  n,  we  can  by  a  proper  choice 
of  the  coefficients  of  this  polynomial,  and  of  the  constant  &?, ,  cause 

the  terms  in  — ,  — j" .  •   •  •  ,   ~~;r-i~i  of  ^h^  product  to  disappear.     Equa- 
tion (135)  shows  that  P„{x)  is  such  a  polynomial. 

As  we  have  assumed  Qj(_x)  to  have  any  one  whatever  of  its  de- 
terminations, such  that  (120)  admits  as  a  solution  an  integral  poly- 
nomial P„{x),  the  theorem  just  proved  holds  true  for  all  polynomials 
of  degree  n  which  satisfy  the  differential  equation 

whatever  be  the  choice  of  the  polynomial   Q.i{x).     Heine  has  given 
an  analogous  theorem  for  polynomials  satisfying  Lame's  equation. 

The  previous  results  are  readily  extended  to  the  case  where  the 
differential  equation  has  p  finite  critical  points  instead  of  only  3  ; 
for  these   generalized    results    the    reader    is,  however,  referred  to 


EQUATIONS  OF  SECOND   ORDER— POLYNOMIAL  INTEGRALS.     211 

Humbert's  paper.  In  concluding  his  paper  Humbert  gives,  without 
going  into  details,  a  theorem  for  the  general  case  in  which  the 
constants 


>w. ,     A', 


//p 


are  not  all  positive ;  for  this  also  the  reader  is  referred  to  the  paper, 


P 


I 


L 


CHAPTER   VII. 

ON   THE   LINEAR    DIFFERENTIAL   EQUATION   WHICH   ADMITS   THE 
HYPERGEOMETRIC   SERIES   AS   AN   INTEGRAL.  ^ 
By  M.  Edouaru  Goursat. 

I.  The  hypergeometric  series  F{a,  fi,  y,  ;tr),  considered  as  a  func- 
tion of  the  fourth  element  x,  is  only  defined  for  values  of  this 
variable  which  have  moduli  less  than  unity.  In  order  to  define  it 
for  any  value  of  x,  it  should  be  regarded  as  a  particular  integral  of  a 
linear  differential  equation  of  the  second  order ;  the  problem  then 
comes  to  finding  what  this  integral  becomes  when  the  path  described 
by  the  variable  ends  at  a  point  situated  07itside  the  circle  of  radius 
unity  and  having  the  origin  as  centre.  By  a  generalization  which 
immediately  presents  itself,  one  is  then  led  to  propose  the  same 
problem  for  any  integral  whatever  of  the  equation,  the  path  followed 
by  the  variable  being  simply  subjected  to  the  condition  of  not 
passing  through  any  critical  point  of  the  equation. 

This  celebrated  equation,  studied  by  Gauss*  and  Kummer,f  ad- 
mits, as  the  latter  has  shown,  twenty-four  integrals  of  the  form 

^/(i  _  xyF{a',  ft',  y',  z), 

where  z  is  one  of  the  variables 

I  I  X  X  —    I 


I  —  X 


provided  none  of  the  numbers  y,  y  —  a  —  ft,  ft  —  a  \s  an  integer. 
In  the  same  memoir  Kummer  gives  the  linear  relations  with  con- 
stant coefficients  which  connect  any  three  of  the  integrals;  but,  as 
he  only  considers  the  case  of  a  real  variable,  the  formulae  which  he 
obtains  present  some  difficulties  when  we  wish  to  pass  to  their 
applications,  the  more   so  as  he  does  not  always  sufficiently  define 


*  Collected  Works,  vol.  iii.  p.  207.  f  Crelle,  vol.  xv.  p.  3g. 

212 


/:   I  / 


GOURSAT:    II  YP F.KGHOMETRIC   SERIES.  213 

the  sense  of  his  integrals.  Suppose,  in  fact,  that  we  wish  to  pass 
from  a  real  value  of  x^  positive  and  less  than  unity,  to  a  real  value 
of  X,  positive  and  greater  than  unity  ;  we  cannot  do  this  by  giving 
X  a  series  of  values  all  of  which  are  real,  because  we  may  not  pass 
through  the  point  -r  =  i,  which  is  a  ctitical  point  of  the  differential 
equation.  It  is  necessary  then  to  give  to  .r  a  series  of  imaginary 
values,  which  can  be  done  in  an  infinite  number  of  ways,  and,  the 
final  value  of  the  function  depending  on  the  law  of  succession  of  the 
different  values  of  the  variable,  we  see  at  once  that,  for  the  proposed 
object,  it  is  necessary  to  introduce  the  consideration  of  imaginary 
values  in  the  differential  equation. 

The  general  theory  of  this  equation,  when  no  restriction  is  im- 
posed on  the  value  of  the  variable,  has  not,  so  far  as  the  author 
knows,  up  to  this  time  been  treated  completely.  Tannery,*  how- 
ever, has,  by  employing  Fuchs's  method  for  linear  differential  equa- 
tions, shown  that  we  can  find  all  of  Kummer's  integrals  ;  in  another 
memoir  f  he  has  determined  the  linear  relations  between  the  inte- 
grals for  a  numerical  example  previously  studied  by  Fuchs.  The 
results  obtained  are  in  accord  with  those  deduced  from  the  general 
case. 

2.  In  Part  First  the  author  develops  first  a  method,  due  to 
Jacobi/"for  finding  the  twenty-four  integrals  of  the  differential  equa- 
tion when  no  one  of  the  numbers  y,y  —  a  —fS,  y5  —  «  is  an  integer; 
then,  by  applying  Cauchy's  theorem  to  the  definite  integrals  which 
represent  the  hypergeometric  series,  the  relations  between  the  inte- 
grals themselves  are  obtained.  If  among  the  numbers  y,  y  —  a—ft, 
f-i  —  a  there  are  one  or  more  integers,  there  will  be  a  change  of  the 
analytical  form  of  certain  of  the  integrals ;  the  new  integrals  are 
found  by  a  well-known  process,  one  which  occurs  particularly  often 
in  the  theory  of  differential  equations.  Part  First  closes  by  seek- 
ing the  conditions  to  be  satisfied  by  the  elements  a,  ft,  y  in  order 
that  the  general  integral  may  be  algebraic.  We  are  so  conducted  to 
a  geometrical  representation  identical  with  one  already  found  by 
Schwarz  \.  in  starting  from  totally  different  principles. 


*  Annates  de  V Ecole  normale  supe'rieure,  2^  serie,  t,  iv.  p.  113. 
f  lb.,  t.  viii.  p.  169. 
,-: :        ^Qx€i\^,  vol.  Ivi.   p.  149. 


214  LINEAR  DIFFERENTIAL   EQUATIONS. 

Part  Second  is  devoted  to  the  investigation  of  the  transforma- 
tions which  the  hypergeometric  series  will  admit  of  when  of  the 
three  elements  a,  yS,  y,  two  only,  or  even  one  only,  remain  arbitrary. 
We  arrive  at  all  the  transformations,  of  a  very  general  form,  and 
containing  all  the  transformations  indicated  by  Kummer,  and  a  cer- 
tain number  of  others  believed  to  be  new.  Finally,  a  number  of  the 
numerous  formulae  which  can  be  deduced  are  given. 

3.  Letting 

V  —  ifi-  "(i  —  liy  -  ^  -  '(i  —  xii)-  % 
we  know,  according  to  Euler,  that  if  the  integral 

So    ^^^ 

is  capable  of  representing  a  function  {i.e.,  a  function  of  x),  then  this 
function,  say  j, 

satisfies  the  differential  equation 

(I)  x{i  -  x)^^  +  [;.  -(^+  /i+  i>]^  -  a§y  =  O. 

In  order  to  demonstrate  this  it  is  only  necessary  to  replace  in  the 
first  member  of  this  equation  y,  -^  ,  -^,  by  their  values.  Ihe 
result  of  this  substitution  is  evidently 

Now  by  a  very  simple  process  we  find  that  the  function  under  the 
sign  of  integration  is 


_  ^ 

du 


■u{\-  u)^/ 
I  —  xu 


GOURSAT:  HYPERGEOMETRIC  SERIES. 
The  result  of  the  substitution  is  then 

"z^(  I  —  ii)~\ ' 


215 


u(  I  —  U)\ ' 

—    «       -^ y 

_  I   —  Xli  Ja 


which,  under  the  supposition  that  the  definite  integral  has  a  mean- 
ing, is  identically  zero. 

Consider  now  the  more  general  integral 


7 


-r 


Vdu, 


and  let  us  assume  that  this  integral  has  a  meaning,  that  is,  defines  a 
function,  whatever  particular  values  we  assign  to  the  limits  g  and  //, 
where  ^  and  //  are  constants.  If  we  substitute  this  value  of  j/  in  the 
first  member  of  equation  (i),  we  have  as  the  result  of  the  substitu- 
tion 


'ti{i  —  ?/) 


V 


In  order  that  this  may  vanish  it  is  only  necessary  to  give  to  g  and  k 
one  of  the  values  O,  i,  ±  00 .  It  follows  then  that  each  of  the  three 
integrals. 


y 


—  I    Vdu,       y  =   j     Vdu,       y 


s: 


Vdu, 


if  they  have  a  meaning,  satisfies  the  above  differential  equation. 
Let  us  consider  now,  following  Jacobi,  the  definite  integral 


=  /""  Vdu, 


where  g  and  e  are  constants,  and  substitute  as  before  in  the  first 
member  of  equation  (i)  the  values  of  j,  y' ,  and  y" .  The  result  of 
these  substitutions  is  readily  found  to  be 


-{y-  ft-  i)e\i  -  e)' 


-^ag\\  -  g)-i-\\  -  xg)---^. 


2l6  LINEAR  DIFFERENTIAL   EQUATIONS. 

If  we  take  e  =  i  and  give  g  one  of  the  values  o,  i,  -f-  oo ,  this  result 
will  be  zero  provided  i  —  a  has  its  real  part  positive,  and  also  pro- 
vided that  the  product  u\\  —  tiy-^{\  —xu)-'^~^  vanishes  for  the 

I 

limit  u  =  ^;  t.^.,  provided  that  the  integral    /  "^  Vi/zt  has  a  meaning. 

We  have  now,  if  all  of  the  above  conditions  are  satisfied,  six 
functions,  defined  by  definite  integrals,  which  satisfy  the  given  dif- 
ferential equation.  The  following  is  a  table  of  these  six  integrals 
together  with  the  conditions  which  must  be  satisfied  by  the  quanti- 
ties a,  /?,  y  in  order  that  the  integrals  may  have  a  meaning. 

The  inequality  ^  >  o  indicates  simply  that  the  real  part  of  A  is 
positive. 

(i)    y  =    I    Vdu        .  .  .     y5  >  o,  y  —  I3>  o; 

n—  CO 

(2)  y  —    I        Vdu   ...     /?  >  O,  or+I— ;(/>o; 

(3)  y  '=^  J,         Vdu...     y  —  ^>0,  a-^l—y>o; 

I 
{4)    y  =    f^  Vdu       .../?>  O,  l-a>0] 

I 

(5)    y  =  J^'Vdu       ...     y  —  /3>0,  I  —  or  >  o  ; 

Vdu    ...     a-\-  I  —  y>  O,       I  —  a>  O. 

X 

4.  Before  going  farther  it  is  desirable  to  define  exactly  those  values 
of  the  function  F  which  we  can  take  in  the  above  integrals,  and  also 
to  examine  more  closely  the  properties  of  the  functions  represented 
by  these  definite  integrals. 

5.  The  integral 


y 


=    f  21^-  '(i  —  u)y  -  ^  -  '(i  —  xu)- 


"du 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


217 


has  a  meaning  provided  that  the  real  parts  of  (5  and  y  —  fi  are  posi- 
tive, and  provided  in  addition  to  this  that  x  has  no  real  value  greater 
than  unity.  We  will  take  this  integral 
along  the  straight  line  o  —  i  (Fig.  i) ;  we 
will  also  take  o  for  the  value  of  the  argu- 
ments of  II  and  I  —  u,  and  for  the  argu- 
ment of  I  —  xu  we  will  take  that  value 
which  reduces  to  zero  when  //  =  o. 

The  function  y  thus  defined  is  a  uni- 
form function  of  x  in  the  entire  plane  if 
only  the    path  described  by  the  variable 

does  not  cut  the  line  i [-  00  .  Pig  ,_ 

Suppose    now    the    point  x    describes    a 

closed  curve  which  does  not  cut  this  line;  then  the  point  X7i,  corre^ 
spending  to  a  value  of  u  lying  between  o  and  i,  will  describe  a  closed 
curve  homothetic  to  the  preceding  one,  and  leaving  the  point  i  out- 
side. The  argument  of  i  — ;r?^  will  then  resume  its  initial  value; 
the  same  thing  will  be  true  of  the  element  of  the  integral  which 
corresponds  to  this  value  of  71,  and  consequently  for  the  integral 
itself. 

It  follows  from  what  precedes  that,  inside  a  circle  of  radius  unity 
and  having  the  origin  as  centre,  the  function  y  can  be  developed  in 
a  series  going  according  to  ascending  powers  of  the  variable  x. 

The  coefficient  of  x'"  in  this  development  will  be 


d'^y 
dx'" 


that  is. 


a{a  -\-  i)  .  .   .  {a  -\-  m  —  i)    /" 
I  .  2  .  3   .   .   .  ;;^  t/ o 


1.2.3.   •  •  ^'^  ' 

z/^+"'  -  '(i  —  7i)y-^-''du, 


or 


a{a  -^  l)  .  .   .  {a  ^  m  —  i)     r(yS  +  m)r{y  —  0) 
P  I  .  2  .  3   .  .  .  ;;z 

or,  finally, 


.  m  r{y  -{-  m)  ' 


1.2.3 


•  r(r  ^-  0  •  •  •  (r  +  ^«  -  0 


nr) 


2l8  LINEAR   DIFFERENTIAL   EQUATIONS. 

We  have  then,  for  every  value  of  x  for  which  \  x  \  <  i, 

The  preceding  integral  can   take  three  other  forms,  which  can  be 
at  once  obtained  by  the  following  transformations  due  to  Jacobi : 

«  =  I  —  t;; 

V 


u  = 


I  —  X  ~[-  vx' 

I  —  : 

I  —  vx' 


By  the  first  of  these  substitutions  the  integral   /    Vdu  is  changed 
into 

(i  -  x)--^£vy  -  ^  -  '(i  -  vf  -  '(i  -       ""    -t)  "^ dv. 

This  integral  is  taken  along  the  same  path  as  the  one  already  con- 
sidered.    If  we  make  the  same  suppositions  as  above  concerning  the 

x 
arguments  of  v,   \  —  v,    i v,  it  will  be  necessary,  in  order 

that  the  two  integrals  may  be  identical,  to  take  for  argument  of 
I  —  X  that  one  which  is  zero  when  x  =  o.     We  have  then 

J^  u^-'{i  —  ny  -y  -  '{i  —  xi()  -  "^du 
Make  now  the  second  transformation, 


I  —  X  ~\-  vx' 
the  integral  becomes 


{i-^)-'S^''-K^-t')'-'-i^  --zr,') 


X        \-(v-i^ 
dv 


GOURSAT:   HYPERGEOMETRIC  SERIES.  2I9^ 

This  integral  is  taken  along  an  arc  of  circle  oM\  ;  it  is  easy  to  see 

that  the  area  comprised  between  this  circle  and  the  line  o  —  i  does 

X I 

not  contain  the  point ,  which  is  a  critical  point  for  the  new 

X  —    I 

corresponds  to  z/  =  oo  ;  the  point,  say  A,  which   represents  

must  thus  belong  to  the  circumference  of  which  oM\  is  an  arc.  We 
can  consequently  take  the  integral  along  the  straight  line  o —  i, 
and,  making  the  same  conventions  concerning  the  arguments  as. 
before,  we  can  write 

The  transformation 

I  —  V 

u  = 

I  —  vx 

gives  in  like  manner 

J^    Vdu  =  (i  —  ;r)v  -  »  -  P^  2;Y  -  P  -  i(i  _  ^f  -  '  (i  —  vx)-  (v  -  "Wz;. 

Each  of  the  new  integrals  can  be  developed  in  a  convergent  series 
for  values  of  the  variable  lying  between  certain  limits.  Thus,  for 
values  of  x  such  that 

I  ^  I  <  I, 
we  have 

(l    —  x)y-'^-^J    Vl-^-\\   —  vy  -  ^(l    —  vx)-  (V  -  «Wz; 

r(^)r(r-/?), 


r{r) 

and  when 


(I  -x)y---PF{r-a,  y-fi,  y,  x); 


<  I, 

X  —  I 


220  LINEAR   DIFFERENTIAL   EQUATIONS. 

we  have 

dv 


For  brevity  we  will  denote  by  C^ ,  C^,  the  circles  of  radius  unity 
having  the  points  o  and  i  respectively  as  centres.  Further,  denote 
by#-£'„  and  ii,  the  portions  of  the  plane  limited  by  the  common  chord 
(produced  indefinitely)  of  the  circles  C^  and  Cj  ,-£"„  containing  the 
point  o  and  B,  the  point  i.  It  results  from  the  previous  considera- 
tions that  the  proposed  differential  equation  admits  an  integral 
which  is  uniform  throughout  the  plane,  provided  only  the  variable  is 
subjected  to  the  condition  of  not  crossing  the  line  i \-  oo.  De- 
note this  particular  integral  by  0, .     In  what  follows  we  will  suppose 

additionally  that  the  variable  does  not  cross  the  line  —  co o, 

though,  as  will  be  seen,  this  restriction  is  not  really  necessary. 
In  the  circle  C,  we  have 

0^  =  F{a,  /3,  y,  x)  =  (i  -  x)"  "'-^Fiy  -a,  y  -  ^,y,  x), 

and  in  the  space  E^ 

<P^  =  {i-x)- '^F[a,  y-  ft,y,  — ^-) 

:=,(^l_x)-^F\y-a,  ^,y,-^J, 

the  argument  of  i  —  ;ir  being  supposed  to  lie  between  —  n  and  +  n. 
The  previous  results  have  only  been  established  on  the  hypothesis 
that  the  real  parts  of  /?  and  ;/  —  /?  are  positive,  but  it  is  clear  that  these 
results  will  still  subsist,  provided  only  that  y  is  not  zero  or  a  nega- 
tive integer.     Suppose  we  wish  to  verify  that  one  of  the  preceding 


G  OUR  SAT:   HYPERGEOMETRIC  SERIES.  221 

functions  satisfies  equation  (i);  the  sign  of  the  quantities  /?,  ^  —  /? 
will  not  enter  into  the  calculation,  and  consequently  the  verification 
will  be  the  same  in  all  four  cases.  The  same  is  true  if  we  wished  to 
verify  that  two  of  these  functions  are  equal  for  values  of  x  which 
make  the  two  series  convergent.  This  remark  is  made  once  for 
all  in  order  not  to  have  to  refer  to  it  in  analogous  cases.  If  the  real 
parts  of  §  and  y  —  §  are  positive,  we  have 

6.  The  integral 

/-    50 

t 

has  a  sense  if  only  the  real  parts  of  ^  and  a  -\-  i  —  y  are  positive,, 
and  if  x  has  no  real  negative  value.  It  is  readily  demonstrable  that 
the  function  y  is  uniform  in  the  entire  plane,  provided  only  that  the 

path  described  by  the  variable  x  does  not  cut  the  line —  oo O; 

the   same   is   true  if  the  path  described  by  the  variable  is    further 

restricted  to  not  cut  the  line   i 1-  oo .     In  order  to  exactly 

define  the  sense  of  this  integral  it  remains  to  choose  the  arguments, 
of  11,  I  —  u,  and  i  —  xn  ;  for  the  argument  of  i  —  ti  we  will  take  o, 
for  the  argument  of  i  —  xn  that  which  reduces  to  zero  for  m  =  o; 
but  for  u  we  will  take  the  argument  ±  tt.  Suppose  first  we  take 
arg.  u  =:  -\-  TT,  and  make  the  change  of  variable  defined  by 


u  = 


du  =  (—  i) 


V  —  I        ^        '  I  —  a 
dv 


(I  -vf^ 

V  varies  from  o  to  i,  and  u  from  o  to  —  oo : 

I 


I 

—  u 

— 

I 

> 
—  V 

I  - 

-  xu 

= 

I 

—  v{\ 

— 

^) 

I  — 

V 

222  LINEAR  DIFFERENTIAL  EQUATIONS. 

If  we  take  o  for  the  arg.  of  v  and  of  i  —  v,  we  ought  to  take  n  for 
the  arg.  of  ( —  i),  and  the  arg.  of  \  —  v{\  —  x)  will  be  zero  for  v^o. 
By  this  change  of  the  variable  the  integral  becomes 

^^^i  j    ^p  -  i^j   _  ^jy  -  v|-j  _  ^(-j  _  ^^-j  -  a^2/. 

In  this  last  integral  the  arguments  of  z',  i  —  v,  i  —  v{i  —  x)  have  the 
same  sense  as  in  the  integral  already  studied.  We  can  therefore 
apply  the  preceding  transformations  to  this  new  integral,  and  so 
conclude  that,  subject  to  the  conditions  indicated  above,  while 
.  a  -\-  jS  -\-  I  —  y  is  neither  zero  nor  a  negative  integer,  the  differ- 
ential equation  (i)  admits  a  new  inlegral,  <p^,  which  is  holomorphic 
in  the  entire  plane. 

In  the  circle  C",  we  have 

0,  =  F{a,  p,  a  -\-  /3  -\-  I  —  y,   I  —  x) 

=  X'-  yF{a  -^l  —  y,  /3-\-l—  y,  a-\-/3-\-l—  y,  I  —  x), 

and  in  the  space  £^ 

<P,  =  x-''F[a,   a-\-l—  y,  a-{-/3-]-i-y,    ^— ) 

=  x-^F  [^,  J3+1  -y,  a-\-^+i  -  y,   ^-), 

arg.  X  lying  between  —  tt  and  -j-  tt. 

When  /3  and  a  -{-  i  —  y  have  their  real  parts  positive  the  func- 
tion 0j  can  be  represented  by  a  definite  integral,  viz., 

supposing  in  the  integral  arg.  u  =  -\-  tt.  If  we  take  arg.  u  =  —  tt, 
we  will  have 


GO  URSA  T :   HYPERGEOMETRIC  SERIES.  223 

7.  The  integral 

/+00 
n»  -  1(1  _  u)y-^-  '(i  —  xii)-°-du 


iTias  a  sense  provided  the  real  parts  oi  y  —  yS  and  o"  -|-  i  —  y  are 
;  positive  and  x  has  no  real  value  lying  between  zero  and  unity.  It 
is  easy  to  see  that  if  the  variable  describes  a  closed  path  enclosing 
the  line  o i,  the  function  y  takes  its  original  value  multi- 
plied by  e^  ^'"*';  the  function  can  therefore  be  rendered  uniform  if 
"we  make  the  convention  that  the  path  of  the  variable  must  not  cut 

the  line  o \-  co  .     As  to  the  path  followed  by  the  variable, 

then,  this  new  integral  presents  an  essential  difference  from  the  two 
integrals  already  examined,  which  arises  from  the  fact  that  we  can- 
not pass  from  the  upper  half  of  the  plane  to  the  lower  half,  or  con- 
versely, by  crossing  the  line  o i.      The  integral  ceasing  to 

have  a  meaning  for  a  point  of  this  line,  there  is  nothing  to  indicate 
that  the  analytical  continuation  of  the  function  would  be  represented 
by  the  same  symbol  after  crossing  the  line;  in  fact,  we  shall  see  that 
the  same  symbol  will  not  answer  after  the  crossing. 

In  order  to  definitely  arrive  at  the  sense  of  the  integral  we  will 
take  o  for  the  arg.  of  ?/,  and  for  arg.  (i  —  xii)  that  which  is  zero  for 
^  =  o,  and  which  varies  continuously  when  the  variable  describes 
the  positive  part  of  the  axis  of  x.     As  \.o  \  —  u,  we  will  take 

arg.  (i  —  ii)~  ±  Tt. 

Suppose  arg.  (i  —  71)  =:  —  n  \  make 

I         ,         —  dv 
u  =■  — ,     ail  ^=^  - — 5 — , 

V  V 

'If  we  take  arg.  v  =  o  and  arg.  (i  —  t;)  =  o,  and  for  arg.  [  i  —  —  ]  that 

i  which  is  zero  when  ^  =:  o,  we  ought  to  take  arg.  (—  i)  =  —  tt,  and 
for  arg.  x  a  value  lying  between  —  tt  and  -(-  Tt. 


224  LINEAR  DIFFERENTIAL   EQUATIONS. 

Vdu  now  becomes 

^i +p  -  Y)T/(_  x)-  '^J^v'^  -  y{i  —  i)y  -  ^  -  '^i  —  -  j 


dv. 


I 


This  new  integral  is  of  the  same  form  as  the  one  first  studied,  and 
on  applying  to  it  the  same  transformations  we  arrive  at  the  follow- 
ing results: 

Whenever  a-\-  i  —  /?  is  neither  zero  nor  a  negative  integer,  the 
differential  equation  (i)  admits  as  an  integral  a  function  0^  which  is 
uniform  throughout  the  plane,  provided  the  path  described  by  the    \ 

variable  does  not  cut  the  line  o 1- oo.     This  function  can     ! 

be  developed  in  a  series  as  follows :  Outside  the  circle  C^  we  have 

03  =  {-  x)--F[a,  ar+  I  -;/,  a+l  -  A  -^j 

^  (_  ^)P-v(i  _  x)y-'^-^F[l-l3,  y-ft,  a-\-  I- A^)- 
Outside  the  circle  C,  we  have 

(t>,  =  {\  -  x)-''F\oi,  y-  ft,  «  +  I  -  /?,   Y^ 

=  {—xy-y{i  —x)y-''-'F[a-{-l—y,  1—/3,  «+i— ^,  y— - 

We  suppose  arg.  (—  x),  as  also  arg.  (i  —  x),  to  lie  between  —  rr  and 
+  TT.  If  the  real  parts  oi  y  —  ft  and  of  «  +  i  —  ;k  are  positive,  we 
shall  have 

/'+~  V  /  ,«     .  -ria-^-l—  y)r{y— ft)^ 

^   ,,^-.(1  -u)y-^-\i  -xii)-  '^du  =  e^^+^-y^-  -r{a-}-i-ft)         "^^ 

on  assuming  arg.  (i  —  u)  =  —  n.  If  arg.  (i  —  ?/)=+  n,  we  shall 
have 

J  u^-^{x-ii)y-^-\x-x^tY<^du  =  .(v-^-x).. 7^^_^V-^)       ^-' 


GOURSAT:   HYPERGEOMETKIC  SERIES.  22$ 

8.  We  can  study  in  the  same  way  the  remaining  three  integrals  : 


/    Vdu,         /    Vdu,  /       Vdu. 

t/o  t/ 1  t/ 1 


As  this  study  involves  no  difficulties  it  will  be  sufficient  merely  to 
give  a  summary  of  the  results. 
The  integral 


y 


I  11^  -  '(i  —  u)y  -  ^  -  '(i  —  xii) -"du 


has  a  sense  if  /?  and  i  —  a  have  their  real  parts  positive  and  if  x 
has  no  real  value  between  zero  and  unity.  If  we  subject  the  path 
described  by  the  variable  to  the  condition  of  not  cutting  the  line 
o \-  00  ,  the  function  J/  will  be  uniform  throughout  the  plane. 

This  integral  can  be  thrown  into  the  first  form  by  making  ti  =  — ; 

we  have  then 


=  ^±t3/(_  x)-^  J^  v^~'{i  —  v)-''[i  —  -j     dv. 

In  the  first  integral  we  take  for  arguments  of  \  —  it  and  i  —  xu 
those  which  vanish  for  ii  =  O.  As  to  the  argument  of  ii,  denoting 
by  on  the  argument  of  ( —  x'),  we  will  have  for  the  coefficient  of  the 
second  integral  f±"^',  according  as  we  take  for  this  argument  the 
values  (—  &9  ±  7t).  We  have  now  for  our  differential  equation  a  new 
integral,  0, ,  which,  like  the  preceding  ones,  can  be  developed  in  a 
series.     Outside  the  circle  C„  we  have 

0,  =  (-  xY^F\^  -^ri-y,  ft,  13+1-  a,~ 

=  (—  xY  -y{l  —  x)y-°--^F[l  —  a,  y  —  a,  ft  -\-l  —  a,  - 


k 


226  LINEAR  DIFFERENTIAL  EQUATIONS. 

and  outside  the  circle  C^  we  have 

0,  =  (I  -  x)-^F[fi,  y-a,  ftJ^i-a,  ^-^) 

the  arguments  of  ( —  x)  and  (i  —  x)  lying  between  —  tt  and  -{-  n. 
For  properly  chosen  values  of  yS  and  i  —  a  we  have 

f'n^-^l  -  H)y-  ^-'(l  -X2l)-  '^du  =  e^  npi£{^)£{l_ZZ_^)  0^  ^ 

9.  The  integral 

/+00 
2^^-i(i  _  /^)v-P-i(i  _  xii)-'^du 

X 

has  a  sense  if  the  real  parts  of  i  —  a  and  «  -}-  i  —  ;j/  are  positive, 
and  if  x  has  no  real  value  greater  than  unity ;  further,  the  function 
y  will  be  holomorphic  if  the  path  described  by  the  variable  does  not 

cut  either  of  the  infinite  lines  —  00 o,   i -|-  00. 

This  integral  can  be  put  into  the  first  form  by  the  transformation 

2/  =  —  ,  and  we  then  have 

XV 

u^  -  '(i  —  uy  -^-\\  —  xu)  -  "^du 


:^    ^±  7rz(v-P-i)±T;a^i  "7     /     ^/"-^(l    —  ■Z^)-"(l   —  XZ>)y  ~  ^  ~ 


'dv. 


The  arguments  of  u  and  i  —  u  are  fixed  by  the  continuity  by  sup- 
posing that  we  start  from  the  origin  with  the  argument  o  and  de- 
scribe the   infinite   radius  oL  passing  through  the  point  -•    As  to 

I  —  xu,  we  can  take  arg.  (i  —  xii)  =  ±  tt.  In  the  preceding  for- 
mula we  will  take  the  sign  -f-  or  the  sign  —  before  jnXy  —  /5  —  i) 
according  as  the  point  represented  by  x  is  in  the  upper  or  in  the 
lower  half  of  the  plane,  and  the  sign  +  or  the  sign  —  before  jtai 
according  as  we  take  arg.  (i  —  xu)  =  -{-  tt  or  —  n. 


GOURSAT:   HYPERGEOMETRIC  SERIES.  22/ 

While  2  —  y  vs,  neither  zero  nor  a  negative  integer,  the  differ- 
ential equation  (i)  admits  a  new  integral,  0^,,  which  is  uniform  under 
the  condition  enunciated  above  as  to  the  path  of  the  variable. 

In  the  circle  C^  we  have 

0^  =  X-  -yF{l3  -\-  I  -  y,   a  -{-  I  -  y,   2  -  y,  x) 

=  ;r'-v(i  —x)y-''-^F{l  —a,    I  —  J3,    2  —  y,   x), 
and  in  the  space  £^ 

0,  =  x^-y{i  -x)y---^F[a+l  -y,l  -  ^,  2  -  y, ——) 

\  X  —  1/ 

=  x^-y{l  -x)y-^-  ^F[^fJ  ^i-y,i-a,2-y,  ~^), 

the  arguments  of  x  and  i  —  x  lying  between  —  rr  and  -{-  n.     For 
properly  chosen  values  of  a^  /?,  y  we  will  have 


dii 

^    ,       .      ,^      .r(a'+  I  -  v)r(l  -  a) 

r{2-y)  "^^ 

10.  The  integral 
I 

y  =  I    71^ -'{i  —  ti)y-^-'{i  —  xii) - °- du, 

which  has  a  sense  provided  the  real  parts  of  i  —  arand  y  —  ft  are 
positive,  and  x  has  no  real  negative  value,  is  holomorphic  under  the 
same  conditions  as  the  preceding  one,  and  can  be  thrown  into  the 
first  form  by  the  transformation 

I  —  X       . 
u  = V  4-  I, 

X 

giving  then 

I 

I    u^--'{i —u)y-^-^{i  —xuY'^dii 
—  e±^i^i-^-^)x^-y{\—x)y-''-^   C  vy-^-^{\—i^-''{\  —  ^^^-v\^-^dv. 


228  LINEAR   DIFFERENTIAL   EQUATIONS. 

The  arguments  of  ?/  and  i  —  xii  are  defined  by  the  continuity;  viz,,, 
we  start  from  the  origin  with  the  initial  value  o,  and  describe  the 
straight  line  O 1  ;  then  starting  from  the  point  ^  =  i,  we  will 

describe  the  straight  line  joining  this  point  to  the  point  —  with  per- 
fectly determinate  values  of  these  arguments.  In  the  case  oi  \  —  u 
there  is,  however,  some  ambiguity.  In  the  preceding  formula  we 
must  take  the  sign  -j-  or  the  sign  —  before  7ii{y  —  fi  —  i)  accord- 
ing as  we  take 

arg.  (i  —  II)  =  arg.  (i  —  x)  —  arg.  ;ir  ±  tt. 

While  y  -\-  \  —  a  —  yS  is  neither  zero  nor  a  negative  integer,  we  will' 
have  a  new  integral,  0^ ,  of  the  differential  equation  (i),  which  will  be 
uniform  under  the  same  conditions  as  the  preceding  one.. 
In  the  circle  C^  we  have 

0,  =  (l  —  x)y-''-^F{j  -a,y  —  ft,y^\-a-ft,  \  -  x) 
—  x^-y{\  —  x)y-''-^F{l  —  a,  1  —  J3,  y  -\-  I  —  a  —  /3,  I  —  x),. 

and  in  the  space  E^ 

06  =  x''-y{i  —  x)y-''-^F(y  —  a,  \  —  a,  y-\-l  —  a  —  /3, j 

=  x^-y{\  —  x)y-''-^F{y  —  (3,  I  -  /3,  y-\-i  -«-/?,  f-Ili 

the  arguments  of  x  and  i  —  x  lying  between  —  tt  and  -|-  n.  For 
proper  values  of  a,  /?,  y,  we  have 


/ 


^  u^-r  (I  _  iAy-P-^  U-xii)—du  =  ^±-'-(y-^-^)  ^^^^      ^^^^^ ^^  0^. 


II.  In  summing  up  all  that  precedes  we  see  that,  so  long  as  no 
one  of  the  quantities  y,  y  —  a  —  /3,  (3  ~  a  is  an  integer,  the  proposed 
differential  equation  admits  six  particular  integrals  each  of  which,  in 
different  parts  of  the  plane,  can  be  expressed  in  four  different  ways  by^ 
hypergeometric  series :  these  are  Kummer's  twenty-four  integrals. 

We  can  render  these  integrals  uniform  by  imposing  certain  con- 
ditions upon  the  paths  described  by  the  variable ;  thus,  for  the  in- 


GOURSAT:   HYPERGEOMETRIC  SERIES.  229 

tegrals  0, ,  0^ ,  0^ ,  06  these  paths  must  not  cut  either  of  the  lines 

—  00 o,  I \-  00  ;  for  the  integrals  03  and  0^  the  path  of 

the  variable  must  not  cut  the  line  o \-  co.    These  six  integrals 

divide  into  three  groups,  each  group  containing  the  integrals  which 
behave  in  a  simple  manner  in  the  region  of  a  critical  point.  The 
first  group  contains  the  integrals  0,  and  0, ;  the  second  group  con- 
tains 0j  and  05 ;  and  the  third  group  contains  03  and  0, .  Each  of 
these  integrals  is  susceptible  of  being  represented  by  a  definite  in- 
tegral, provided  the  elements  a,  /3,  y  satisfy  certain  conditions.  The 
following  is  a  table  of  the  twenty-four  integrals: 

*"  Tabic  of  Integrals. 

'(i)     F{a,ft,y,x), 
(2)     {i  -  x)y---^F{y  -  a,  y-  (5,y,x\ 


■0. 


'.  ^(3)  {i-x)-^Fia,y-ft,y,-^^, 

(4)  (I  -x)-^F[p,y-a,y,-^^^. 

(1)  F{a,  13,  a  + ft-\-\  -y,l  -  x), 

(2)  X'-  yF{a  +i-y,  f3+i-y,a  +  l3+i  -  y.  I  -  x\ 

^(3)  x-<'F\a,  a^l  —  y,  a-]^  ft+l  -  y,——\, 

(4)  x-^F[p,f3+i-y,a  +  ^-\-i-y,'^y 

(0  {-x)-'^F{a,  ar+  I  -  X,  «+  I  -  A^j. 

(2)  {-xf'y{i-x)y-'^'^F[i  -^,y-j3,a  +  i^^,'^y 

(3)  {i-x)-'^F(^a,y-/3,a^i-^,:^'j, 

(4)  {-xy-y{l-x)y-<^-^F(a+i-y,  1-/3,  a-{-l-/3,  ^^^ 


-><' 


^^ 


230 


<t>,  i 


LINEAR  DIFFERENTIAL  EQUATIONS. 

(2)  {—  xY--i{\  —  xy-^-^Fxi  —  a,  y  —  a,  (i -\-  I—  a,  -1, 

(3)  {,-x)-^F[^fi,y-a,^-\-l-a,-^^, 


(4)     {-xy-y{l-x)y'^-^F\ft+l-y,  i-a,ft-\-l-a,- 


I  —  X 


'  (i)     x'-yF{oi-^i  —  y,   /?+  I  —  y,  2  —  y,  x),    . 
(2)     ;r'-v(i  -x)y-''-^F{l  -a,  1-/3,  2  -  y,  x), 


<P. 


(3)     ;ir'-^(l  —  ;ir)v  -  »  - '/^ Ui' +  I  —  ;K,  1—^2  —  ;/,  :^:^j^ 


(4)      ;t-'-^(l  -^)^-^--i^f/?+I  -7'   I 


^i-,  2  —  y, 


X  —  I 


0. 


(i)  (i  —  x)y-''-^F{:)^  —  a,  y  —  §,  y-\-\  —  a  —  ft,  \  —  x), 

(2)  ;ir'-^(l  -;i')^-"-^^(i  -  «,  I  -  A  7  +  I  -  «  - /?,  i  -  ;ir), 

(3)  ^"-^l  _;ir)v-«-^/r(^^_a',  I-  a,y^i  —  a  -  /3,^~~^^j, 

(4)  ;f^-v(i  -x)y-''-^F(y-/3,  1-/3,  y-\-l  -  a  -  /3 


X 


12,  Between  three  of  these  integrals  there  is  a  linear  relation  with 
constant  coefficients  in  all  that  part  of  the  plane  in  which  the  inte- 
grals are  holomorphic.  If  we  consider  three  of  the  integrals  denoted 
by  0,,  0,,  06,  06,  the  relation  will  be  unique  throughout  the  plane. 
But  this  will  not  be  true  if  we  take  one  of  the  integrals  03  and  0,  in 
combination  with  two  of  the  preceding  ones.  Take,  for  example, 
0, ,  0j ,  03 ,  and  let  M  and  M'  denote  two  points,  one  situated  in  the 
upper  half  of  the  plane,  and  the  other  in  the  lower  half.  There 
exists  no  path  joining  M  and  M'  which  will  not  cut  at  least  one  of 

the  two  lines  —   00 o,  O [-  00  ;  one  at  least,  then,  of  the 

three  functions  represented   in  the  region  of  M  by  0, ,  0^ ,  03  will      f 


GO  URSA  T:   HYPERGEOMETRIC  SERIES. 


231 


not,  after  travelling  such  a  path,  be  represented  in  the  region  of  M' 
by  the  same  symbol. 

The  preceding  remark  is  essential,  and  it  will  be  useful  to  de- 
velop it.  Suppose  E,  E'  (Fig.  2)  two  areas  with  simple  contours  T, 
T\  neither  of  which  encloses  in 
its  interior  a  critical  point  of  a 
differential  equation  of  the  sec- 
ond order ;  suppose  that  be- 
tween these  two  areas  there  is 
a  critical  point,  A,  of  the  dif- 
ferential equation,  and  suppose 
further  that  the  areas  have  in 
common  two  separate  areas,  C 
and  C — that  is,  such  that  we 
cannot  pass  from  a  point  of  the 
area  C  to  a  point  of  C  without 
cutting  at  least  one  of  the  con- 
tours T,  T.  Let  P  and  P'  be 
two  linearly  independent  par- 
ticular integrals  which  are  uni- 
form in  the  area  £",  and  let  Q  and  Q'  denote  two  such  integrals 
in  the  area  E' . 

In  the  common  part  C  we  have  the  relations 


(I) 

and  in  C  we  have 


Q  =    \P    ^   IxP', 
Q'  =  rp  -{-  ix'P'; 

Q  =  X,P  +  ju^P\ 
Q'  =  K'P  +  fx^'P'. 


It  is  easy  to  show  that  the  relations  (I)  and  (II)  must  be  distinct ; 
that  is,  that  we  cannot  have  simultaneously 


^1  =  A,      /^i 


fX, 


\'  =  V,      /./  =  / 


Suppose,   in  fact,  that  these  last  equations  were  satisfied  ;  let  us 
start  from  a  point  m  in  C  with  the  particular  integral  Q  and  de- 


232 


LINEAR  DIFFERENTIAL   EQUATIONS. 


scribe  a  path  situated  inside  the  area  E,  and  so  arrive  at  a  point  m' 
of  C .  All  along  this  path  our  integral  will  be  given  by  \P  -f-  jxP' , 
and  under  our  hypothesis  will,  at  the  point  m' ,  coincide  with  the 
particular  integral  Q.  If  now  we  return  to  the  point  ni  by  a  path 
lying  in  E\  we  will  evidently  arrive  at  this  point  with  the  original 
integral  Q.  We  would  therefore  have  described  a  closed  curve  con- 
taining the  point  A  and  have  returned  the  integral  to  its  original 
value.  The  same  would  be  true  if  we  had  started  out  with  the  par- 
ticular integral  Q .  The  point  A  could  not,  therefore,  be  a  critical 
point  for  the  differential  equation.  It  is  clear  that  in  the  case  under 
consideration  the  areas  C  and  C  coincide  respectively  with  the 
upper  and  lower  halves  of  the  plane,  and  the  contours  T  and  T'  with 

the  lines  —  co o,   i \-  oo  ,  and  o f-  oo  , 

It  is  thus  established  that,  as  they  have  been  defined,  there 
should  exist  between  the  integrals  0, ,  0, ,  03  two  different  linear 
relations  according  as  the  point  x  is  in  the  upper  or  lower  h^lf  of 
the  plane. 

13.  Let  us  assume  the  point  ;r  in  the  upper  half  of  the  plane,  and 

suppose  further  that  the  real 
parts  of  (i,  y  —  /3,  a-\-l— y 
are  positive. 

Describe  around  the  points 
X  =  o  and  X  =  I  two  semi- 
circles ajna'  and  dud'  with  very 
small  radii  r  and  r'  respectively, 
and  around  ;ir  =  o  describe  also 
a  semicircle  LML'  with  a  very 
large  radius  R ;  the  function 
K=  ?^^  ~'(i  —?/)■>' ~  ^  "' (i — ;ir?^)~*will  be  holomorphic  inside  the 
area  bounded  by  these  semicircles  and  the  portions  of  the  straight 
line  L'a' ,  ab,  b'L. 

By  Cauchy's  theorem  we  have 


iVdu  =0, 


where  the  integration  extends  around  the  entire  contour  just  de- 
scribed. 


GOURSAT:   HYPERGEOMETRIC  SERIES.  233 

This  can  be  written  in  the  form 

Vdu  +  /      Vdu  +  /      VdH=  -         Vdu  -         Vdu  -         Vdii. 

If  now  the  radius  R  increase  indefinitely,  and  if  the  radii  r  and  r' 
tend  towards  zero  at  the  same  time,  it  is  easy  to  see  that  each  of 
the  three  integrals  in  the  second  member  of  this  last  equation  tends 
towards  zero.     Take,  for  example,  the  integral 

/     Vdu  —  I     11^-  '(i  —  7i)y-^-  '(i  -  xu)-  ""du  ; 

for  values  of  11  for  which  mod.  u  is  very  great  this  integral  can  be 
written 

{—x)-''l     icy-''-^{i-\-e)du, 

^  LML' 

-where  e  is  an  infinitely  small  quantity.     Write  now 

u  =  Re'^  ; 


then    /     Vdu  becomes 

i{-x)-  «  r R^y  -  -  -  ')'■»(!  +  e)de. 
Let  now  y  —  a  —  \  =  jx  -^  zv ;  then 

y   Vdu  =  i{—  x)-  '^r e^^(^'>  +mi^  +  yi)[i  _|_  e)dd 

Since  by  hypothesis  ju  is  a  real  negative  number,  we  can  take  R  so 
great  that  the  maximum  modulus  of  the  function  under  the  sign  of 
integration  shall  be  less  than  an  assigned  number  7;  the  modulus  of 
the  integral  will  then  be  less  than  7r77[mod.  (—  x)-  »];  that  is  to  say, 
it  can  be  as  small  as  we  please.     We  can  show  in  like  manner  that 

each  of  the  two  integrals   /      Vdu,    f    Vdu  has  zero  for  its  limit. 

We  have  therefore  the  equality 

Vdu  +y  Vdu  +y  Vdu  =  o. 

If  we  take  o  for  the  argument  of  u  and  i  —  u  along  the  path  ad,  it 


234 


LINEAR  DIFFERENTIAL  EQUATIONS. 


is  clear  that  we  must  take  -\-  n  for  the  argument  of  u  along  the  path 
L'a' ,  and  —  n  for  the  argument  oi  \  —  u  along  the  path  h' L.  By- 
referring  then  to  the  definite-integral  expressions  for  the  functions 
0, ,  0j ,  03 ,  we  see  that  the  preceding  equation  gives  the  following; 
relation  connecting  these  functions  : 


(I) 


.  r{fi)r{a  +  I  -  r) 

I\a  +  ^  +    I    _  ;.) 


0,  +  ^('+^ 


y)ni 


r{a+l-  y)r{y  -  ^) 
r{a-\-l  -  ft) 


03 


This  relation  has  been  established  by  supposing  the  real  parts  of 
ft,  y  —  fi,  a-\-  I  —  y  to  be  positive.  In  order  to  demonstrate  that 
the  relation  is  general  it  is  only  necessary  to  employ  the  well-known 
method  of  procedure  which  consists  in  showing  that  if  the  relation 
holds  for  values  oi  ft,  y  —  ft,  a  -\-  i  —  y  comprised  between  certain 
limits,  it  is  still  true  when  we  diminish  one  of  these  values  by  unity. 
We  will  show  first  that  the  relation  holds  whatever  be  the  value  of 
a.  If  (I)  is  demonstrated  for  certain  values  of  a,  ft,  y,  it  will  hold 
for  a-\-  I,  ft,  y ;  we  can  therefore  write,  replacing  0, ,  0^,  03  by  the 
corresponding  series, 


r{a+ft+i-y) 


F{a,  ft,  a  -\-  ft  -\-  I  -  y,   1  -  x) 


r{ft)r{y-ft) 
r{r) 


F{a,  ft,  y,  x) 


' F{a-\-  I,  ft,   a  +  ft-\-  2-y,    \  -  x) 


r{a+l-ft) 


r{oL^2-  ft) 


F\a  +  I,   a  -}-  2  -  ;k,   «'  +  2  -  /?,  ^). 


GOURSAT :   HYPERGEOMETRIC  SERIES.  235, 

Multiply  the  first  of  these  relations  by  {a  —  ^)x  -\-  y  —  2a,  and 
the  second  by  «  (i  —  ;tr),  and  add  the  results.     The  coefficient  of 

nmi^)  will  be 

r{y) 

\{a  -/3)x-{-y  -  2a]F{a,  /3,  y,x)-\-a{l-  x)F{a  +  I,  /?,  y,  x), 

that  is, 

{y  —  a)F{a  —  I,  /?,   y,  x), 

by  a  well-known   formula  in  the  theory  of  hypergeometric  series.. 
Further,  in  the  second  member  of  our  equation  we  shall  have 


X     \\{a-  ft)x-i^y  —  2a\F[a,   a -^  I  —  y,  a -^  \  -  ^,  ^ 

a(a  -\-  I   —  y)(l  —  x)       f  I 

The  quantity  in  -j    [  reduces  to 

{a  —  ft)xF\a  —  I,  oc  —  y,  a  —  ft,  -j, 


and  the  coefficient  oi  F\a—  \,  a  —  y,  a  —  /3,  —j  becomes 

-  e<^+^  -V)--  ^(/^  +  I  -  y)ny  -  /?)       _  _,) 

We  find  in  the  same  way  for  the  first  member  of  our  new  equation^ 
the  value 


236  LINEAR  DIFFERENTIAL   EQUATIONS. 

.aiid  SO  have  finally 

^  ,,.+-,„■■  ^<" -^>^<^-  ^)(-.)-<-'f  („  - 1, .  -  ^, .  -  ^,  1) 

which  is  simply  equation  (I)  with  a  changed  into  a  —  \.  This  rela- 
tion is  therefore  exact  whatever  be  a.  In  the  same  way  we  can 
show  that  if  (I)  holds  for  two  values  of  y  differing  by  unity  it  will 
also  hold  for  a  value  of  y  differing  by  unity  from  the  least  of  the 
preceding  values,  and  consequently  that  it  holds  for  all  values  of  a 
and  y.  It  remains  then  only  to  show  that  /?  can  also  be  arbitrary, 
-which  is  done  by  a  process  entirely  similar  to  the  above  and  which 
tieed  not  be  reproduced  here. 

14.  Formula  (I)  is  then  perfectly  general,  and  we  can  readily  de- 
duce from  it  the  following  formulae  : 

r{a  -\-  p-\-  I  —  y)  r{y) 

^  r(-+i-/?)        ^" 

r{a-]-/3   -\-l  -  y)  r{y) 

^  r(/?  +  i-«)        ^- 

(IV)    ,(.4-.-vw-^(/^+  I  -  y)r{a)     _  r(^  + 1  -r)r(i  -ft) 


GOURSAT:  HYPERGEOMETRFC  SERIES.  21T 

Formula  (II)  is  deduced  from  (I)  by  permuting  a  and  y5;  (III)  is  ob- 
tained from  (I)  by  changing  a  into  y  —  a  and  /5  into  y  —  ft  and 
multiplying  by(i  —  xy-°--^,  and  finally  (IV)  is  obtained  from  (I)  by 
changing  a  into  a-\-  \  —  y,  ft  into  ft -\-  \  —  y,  y  into  2  —  y,  and 
multiplying  by  x^  -  y.  Remark  that  if  we  take,  as  we  have  supposed, 
the  argument  of  x  between  —  tt  and  -(-  tt,  and  so  of  course  for  arg. 
( —  x),  we  shall  have 

X  =  {—  x)e'"' . 

The  relations  (I),  (II),  (III),  (IV)  are  distinct,  and  we  can  deduce 
from  them  all  of  the  linear  relations  which  exist  between  the  six 
integrals.  These  relations  are  twenty  in  number ;  among  them  we 
may  remark  particularly  those  in  which  there  appear  two  functions 
of  the  same  group  which  permit  us  to  pass  from  the  region  of  one 
critical  point  to  that  of  another,  and  which  are  useful  for  the  ijite- 
gration  of  the  differential  equation.  There  are  twelve  relations  of 
this  sort ;  the  remaining  eight  relations  are  between  three  functions, 
of  three  different  groups. 


Upper  Half  of  the  Plane. 

rift)ria  +  .-y)        ri^)ny-ft) 

^^  r{a  +  ft+i-y)    '^^  r{y)  ^' 


,    ^,,+.-,).-  r{a-i-l-y)r{y-ft) 


^^  ria  +  ft-^l-y)"^^-  r{y)  ^' 


.    ,(a+x-.w  r{ft+l-y)r{y-a) 

+  r{ft+i-a)        '^- 


(3)    ,(v-.w.-^(r  -  ^)^(i  -  -)  ^^  _  r{P)r{y  -  ft) 


ny  +  l-a-ft)^'  r{y) 


,(.-«..-^(I-^)^(^) 


r{ft-\-l-a)^'' 


238  LINEAR  DIFFERENTIAL  EQUATIONS. 

W)    '  r(a  +  /)+i-r)'^'  r{2-r)         '^' 

,      ,  T(v  -  ff)r(l  -  fi)  r(a)r(y-  a) 

r{y+i-oi~fi)  r(y) 

<6;    ^^       r(„  +  /s  +  ,-r)'''  = ni^T^ ^• 

+  '  r(/S+.-«)'^" 

<,)    ..■„„.r(.  -  /.)r(,  -  .)     ^  r(,  -  ,)r(,  +  .  -  ,) 

^  r(/i+i-.)        '^^' 

^^  r(r  +  i---/5)    ^~  /^(2-r)  ^' 


r(2-;K)r(r-^-/?)    ,     ,     r{2-  y)r{a-\-ft-y) 

r{a+/3+i-y)r{i-y)  ^  ,  r(^+^+i ->/)r(r- 1) ^ 
<^^>    ^^=r(.+i-;.)r(/5+i-K)^'+  n^)Fo^)  '^^' 

r(;/+i-n--/?)r(i-r)^      r(K+i-a'-^)r(r-i)^ 


GOURSAT:  HYPERGEOMETRIC  SERIES.  239 


r{y)r{fi-a)        r{y)r{a-  fi) 


"^r(i-/^)r(«  +  i-r) 


r{2-y)r{y-a)r(Ji) 
^     ^       ^^  F(l  —  ar)r(;K  —  «) 

"^        r(i-/5)r(r-^) 

r{a-^l-y)r{a) 


240  LINEAR  DIFFERENTIAL   EQUATIONS. 

Lower  Half  of  the  Plane. 

/TV    r-^sir{^)r{a  j^,-y)        r{i3)r{y  -  /?) 

,   .v-:-.v.n^+i-r)r(r-^)  . 
^  r(/?+i-.)        ^- 

^^^  r{y+i-a-^)   ^^-  r{y)  ^' 

^5)  rir+i-a-p)"^'-        r{y)       '^• 


(.y      .a-.w  r{l  -  fi)ny-  a)  r(l  -  ^)r {^  +  ,  -  y) 

,3-,wnr-^)r(/i+i-r)  , 


GOURSAT:   HYPERGEOMETRIC  SERIES. 

^^  rir+i-a-js)"^'-  r{2-r)  "^^ 


241 


r{2  -  y)r{a  -  /?) 


+ 


r(i  -  fs)r{a  +  I  -r) 


^(V-I)7r,  ^^^ 


rrcv    ^  _r(i-r)r(aH-i-^)  , 


_  r(K)r(i  -;.)r(^+i-/?)      ^^^, 
r(2  -  ;/)r(K  -  /?)r(a) 


(16)'    0. 


r(i  -^)r(/^+i_;.)^' 


r{a+^+,-y)r{a-p) 
^  r{a+l-y)r{a)  "^^ 


i  (I  —  a)r{y  —  a) 


(19)'    0= 


r (I  -  ^)r(;/  - /5)r(ar  +  /^  +  i  _ ;,)     ' 

_  />  +  ^  -  y)r{a  +1-/?)    „^^_ 
/^(^  +   I    -  ;K)r(«) 


0. 


7)' 


242  LINEAR  DIFFERENTIAL   EQUATIONS. 

^  >    ^'~    r(i-a)r(r-a)r{a  +  fi+i-y)         ^' 

_  na+l3-y)r((i+.-a) 

r{fi+i-y)r(p)  *■•• 

15.  Formulae  (5)  and  (6)  are  deduced  from  (3)  and  (4)  by  permuting 
a  and  ^.  If  in  (4)  we  change  a  into  y  —  a,  /3  into  y  —  ^,  and  multi- 
ply by  (i  —  x)y-'^-^ ,  we  obtain  (7),  and  from  (7)  derive_  (8)  by  per- 
muting a  and  j3.  Formulae  (9)  and  (10)  are  obtained  by  eliminating 
<p^  between  (2)  and  (3)  and  between  (6)  and  (7)  respectively  ;  solving 
(9)  and  (10)  for  0,  and  0,  respectively,  we  find  (11)  and  (12);  solving 
(i)  and  (2)  for  0j  and  cp^ ,  we  get  (13)  and  (17).  If  in  (13)  we  change 
a  into  a  -\-  I  —  y,  /i  into  ^  -{-  1  —  y,  y  into  2  —  y,  and  multiply  by 
jv^-y  ,  we  obtain  (14),  and  from  these  last  two  we  then  derive  (15) 
and  (16).  Formula  (18)  is  obtained  from  (17)  by  changing  a  into 
y  —  a,  (3  into  y  —  (3,  and  multiplying  by  (i  —  ,r)>'-"-^  ;  then  from 
(17)  and  (18)  we  readily  get  (19)  and  (20). 

As  already  remarked,  (9),  (10),  (11),  and  (12)  hold  throughout 
the  plane.  The  same  is  true  for  (13);  for,  the  function  0,  being 
uniform  in  the  region  of  the  point  x  ^  o,  we  may  suppose  that  the 

path  followed  by  the  variable  cuts  the  line  —  00 o.     But  the 

other  formulae  for  a  point  in  the  lower  half  of  the  plane  will  be  dif- 
ferent ;  the  relation  between  0, ,  0, ,  0g ,  for  example,  will  then  be 

o-'l^i(3)r\a4-  I  -  y) 

_r{ft)ny-ft)  ^,^^^_^^^T{a-^.-y)r{y-Jl 

-         r{y)         ^'^"  i>  +  l-^)        ^- 

This  differs  from  (i)  in  that  n  is  replaced  by  —  n.  All  the  other 
formulae  for  the  lower  part  of  the  plane  are  deduced  from  this  by 
permutations  of  the  letters  just  in  the  same  way  as  in  the  upper  half 
of  the  plane. 

16.  Consider  a  path  of  arbitrary  form  joining  any  two  points  M 
and  M'  of  the  plane,  but  not  passing  through  either  of  the  points  O 
or  I.  If  we  start  from  the  point  71/ with  a  particular  solution  of  the 
differential  equation,  this  solution  will  be  defined  all  along  the  path 


G0UKSA7\-   HYPERGEOMETRIC  SERIES.  243 

•described  by  the  variable,  and  we  will  arrive  at  the  point  M' 
with  a  determinate  integral.  The  preceding  relations  permit  us 
to  find  this  integral  when  the  path  from  M  to  M'  is  given,  and 
when  the  particular  solution  with  which  we  start  from  M  is  given. 
Suppose  we  start  from  a  point  A  corresponding  to  a  real  value  of  x 
and  lying  between  x  =1  o  and  x  =^  i,  and  go  to  a  point  M'  of  the 

plane  by  a  direct  path  which  cuts  neither  of  the  lines  —  00 o, 

-j-  I h  CO.    In  the  region  of  A  the  integral  can  be  represented 

by  C01  +  C'(p^  by  giving  the  constants  C  and  C  proper  values.  If 
the  point  M'  lies  in  the  space  E„ ,  we  will  replace  (p^  by  its  value  in 
terms  of  0,  and  0^ ,  and  will  employ  for  the  effective  calculation  of 
the  function  the  most  convenient  development  of  each  of  these  last 
two  functions.  So  also  if  M'  is  in  the  space  E^ ,  we  will  replace  0, 
by  its  value  in  terms  of  0^  and  0^ .  If  M'  is  outside  the  space  com- 
mon to  the  circles  C^  and  C^ ,  we  can  express  0j  and  0^  by  means  of 
03  and  0^ ,  being  careful,  however,  to  use  different  formulae  according 
as  M'  is  in  the  upper  or  lower  half  of  the  plane. 

Suppose  now  we  start  from  A  and  follow  any  path  up  to  A/', 
provided  the  path  does  not  pass  through  either  of  the  points  x  =  o 
■or  ;ir  =:  I.  Such  a  path  can,  as  we  know,  be  reduced  to  a  series  of 
loops  {laccts)  going  round  the  points  o  and  i,  in  one  sense  or  the 
other,  followed  by  a  straight  path  from  A  to  M'. 


Let  us  start  from  the  point  A  with  a  particular  integral 
•6701  -|-  C'(p^,  and  describe  a  loop  in  the  direct  sense  round  the- 
•critical  point  ;r  =  o.  To  see  how  this  integral  behaves,  we  have 
only  to  replace  0^  by 

r{a^fi+i-y)r{i-y)        r{a  + 13  +  I  -  y)r{r  -  i) 
r(«  +  I  -  y)r{fi  + 1  -  r)  '^'  ^  r{a)r{f5)  "p^  • 

When  the  variable  describes  the  loop,  0,  does  not  change,  but  0^ 
changes  into  r^'  -y^^'^'cp^,  so  that  we  come  back  to  the  point  of  de- 
parture with  the  integral 


244  LINEAR   DIFFERENTIAL   EQUATIONS. 


Ccp,  +  c 


r{a+l-y)r{^+l-y) 


_  r{a-\-i3+i-rW{^-y)  . 
^^     7>+i-x)r(A+i-r)   'J 


or  "  <^,0, +  <^/0., 

where 

c  -c-\-c  r{-  +  ^+i-y)r{i-y)    _ 

So  also  for  a  loop  round  x  =:  i  the  integral  C(p^  -\-  C (p^  changes- 
into  Ci0,  -\-  C^(p^ ,  where 

I  {y-  a)r{y  -  (iy  ^ 

The  sign  -|-  or  —  goes  before  27ri{  y —  a  —  ft)  according  as  the  loop 
is  described  in  the  direct  {i.e.,  positive)  sense  or  in  the  opposite  {i.e.y. 
negative)  sense. 

If  the  variable  describes  several  loops  in  succession,  the  formulae 
will  have  to  be  applied  a  corresponding  number  of  times,  and  we  will 
finally  be  conducted  to  results  similar  to  the  preceding. 

Let  us  take  now  the  general  case  where  the  path  of  the  variable 
joins  any  two  points  of  the  plane.  This  path  can  be  replaced  by  a 
direct  path  going  from  M X.o  A,  followed  by  a  perfectly  determinate 
path  going  from  A  to  M'.  In  order  to  be  conducted  to  the  preced- 
ing case,  it  will  be  sufificient  to  determine  with  what  integral  we 
arrive  at  the  point  A  in  following  the  direct  path  from  M  to  A, 
The  above  method  enables  us  to  do  this  without  difficulty. 

Let  us  consider  as  an  example  the  differential  equation 


y  =  oi 

this  admits  as  an  integral  the  hypergcometric  series- 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


245 


Suppose  we  start  from  the  point  A  with  this  solution  and  describe 
the  closed  curve  ABCDA  (Fig.  5),  surrounding  the  two  points  o 
and  I. 

D 


[ 


This  contour  reduces  to  two  loops  described  in  the  negative 
:sense  around  the  points  x  =  i  and  x  =^  o.  After  describing  the 
first  loop  around  ;tr  =  i  we  return  to  the  starting-point  A  with  the 
integral 


r{y)r{y-a-.  ft) 


+ 


r{y-a)r{y-ft) 


{i-e-^^'-(y-<^-P))F{i,i,i^,  i-x); 


after  describing  the  second  loop  we  will  have  an  integral  which,  in 
the  region  of  A,  can  be  represented  by 

i..  CF{i,  i  1  x)  +  C,F{i,  ^,1^,   I  -  x), 

where 

sin  (y  —  a)7t  sin  {y  —  /?);r 


+ 


sin  y7C  sin  (y  —  a  —P)7t 

r{y)r{y  -a- ft)    _ 


(l   —  ^-2'''(y-«-^))(i   _  ^-2ir/(i-7)). 


c.^ 


r{y-a)r{y-  ft) 
Making  now  a  =  i,  /?  =  i,  ;/  =  |,  we  have 

C  =  e^  -\-2(i  —  e^)=2—  e 


^-27r/(i-Y')/j    ^-2,Ti(y-a-P)\ 


3    , 


i 


246  LINEAR  DIFFERENTIAL  EQUATIONS. 

17.  When  y  is  an  integer,  or  indeed  when  y  —  a  —  (3,  ox  a  —  /?,, 
is  an  integer,  we  have  only  one  integral  in  one  of  the  groups;  in  order 
to  find  a  new  integral  we  employ  the  following  well-known  process. 
Let 

y  =  F{x,  r),  y,  =  F,{x,  r) 

be  two  distinct  integrals  of  the  differential  equation  which  become 
equal  for  a  particular  value,  r  =  r, ,  of  the  constant  r.  We  will  ob- 
tain another  integral  by  seeking  the  limit  for  r  =  r^oi  the  expres- 
sion 

F{x,  r)  —  F^x,  r) 

> 
r-r, 

which  is  also  an  integral  of  the  differential  equation  whatever  be  the 
value  of  r. 

Let  us  suppose  y^  to  be  an  integer ;  we  may  also  assume  y  posi- 
tive ;  for,  if  it  were  negative,  we  could  make  the  transformation 
y  =  x''~'^y^ .  There  are  two  cases  to  be  distinguished  according  as  y 
is  equal  to  unity  or  greater  than  unity. 

Firs^  Case:  y  =  i. — The  two  integrals 

F{a,  ft,  y,  x)     and     x'-yF{a  -\-  i  —  y,  ft-\-  i—  y,  2  —  y,  x) 

become  identical  for  ;^  =  i.  From  what  has  been  said  we  must 
seek  now  the  limit  for  ^  =  i  of  the  expression 

x'-yF{a  -{-  \  —  y,  ft-^l  —  y,   2  —  y,  x)  —  F{a,  ft,  y,  x) 

I  -  y 

\ 
This  limit  is  obviously  equal 

in  which  we  make  y  z=  i.  As  the  function  0j  is  susceptible  of  four 
different  forms,  there  must  equally  be  four  different  forms  for  the. 
new  integral.     Let 

<p,  =  F{a,  ft,  y,  x), 


GOURSAT:  HYPERGEOMETRIC  SERIES.  247 

and  write 

and  denote  by  A^  the  coefficient  of  x""  in  the  development  of 

F{a,  /3,  y,  x). 
We  find  then  readily 

W=  +00 

ip,{x)  =     2    A„,B^x'", 
where 

"        a-^a+I^^a+OT-I^/S^/3+I^ 

If  we  take  0,  =  (i  —  x)y-'^-^F{y  —  a,  y  —  ft,  y,  x),  and  denote  as 
above  by  -^^  the  coefficient  of  the  general  term  in  the  series 
F{y  —  a,  y  —  ft,  y,  x),  we  shall  have  for  ipi{x)  the  new  form 

rt!=-\-co 

tp^{x)  =     2    A,„B,,x'"{i  -  xy-'^-^ 
where 

'"        1—  a^2-a^^7;i-(x    '     1-  ft  ^  2-ft^  '  '  ' 


Each  of  the  two  expressions  for  0,  will  give  a  different  expression 
for  ^j .     Thus  we  find 

tp,{x)  =  -  log  (I  -  x){i  -  x)-'^F[a,   I  -  ft,   I,  -^-) 

\  X  —  il 

+  (i  -,r)-«2  A,M-^-)    , 
«  =  i     .       \x  —  1/ 


A, 


_  a(a-\-  1)  .  .  .(a-{-m—  i)(i  —  ft){2  —  ft)  ...  {m  —  ft) 
~  (1.2...  my  ' 


248  LINEAR  DIFFERENTIAL  EQUATIONS. 

Also, 

V'.W  =  -  log  (I  -  ^)(i  -  x)-^F\p,  I  -  «,  I,  ^^-) 

'« =  °°  /      ar       \  *" 

_  ^(/?  +  I)  .  .  .  (yg  +  ^^  -  0(1  -  ^)(2  -  g)  ...  (;;^  -  6.) 

"•"  (1.2...?/^)^ 

+  -^--2(1  +  1+1+..  .  +  1). 

Whichever  be  the  expression  adopted  for  '/'X-^),  we  will  denote  by  Q 
the  new  integral,  viz., 

^  =  0,  log  ^  +  tp,{x). 

This  new  integral,  like  the  preceding  ones,  will  be  uniform  through- 
out the  plane  provided  the  path  described  by  the  variable  does  not 

cut  either  of  the  lines  —  co o,  i \-  00.     The  method 

employed  for  determining  the  new  integral  enables  us  also  to  find 
the  linear  relations  connecting  it  and  the  integrals  already  known. 
To  fix  the  ideas,  suppose  that  the  sum  a -\-  /3  is  neither  zero  nor  a 
negative  integer,  and  give  to  y  a.  value  differing  but  little  from 
unity.     We  know  the  three  integrals 

F{a,  ^,  y,  x),         x^-yF{a  -\-  i  —  y,  ^  -\-  I  —  y,  2  —  y,  x), 

F{a,  /?,  «  +  /?  +  I  —  ;>/,  I  —x), 

between  which  exists  the  relation 

r{a+/3^l-y)r{l-y)  r{a-\-ft+l-y)r{y-l) 

^'    r{a+i-y)r{ft+i-y)'^''^  r{a)r{/3)  "P-- 


GOURSAT:  HYPERGEOMETRIC  SERIES.  249 

Replace  in  this  relation  0,  by  0i  +  (i  —  y)Qxi  and  let  y  tend  to  the 
value  unity;  0,  and  0^  will  reduce  respectively  to  F{a,  fi,  i,  x)  and 
F{a,  ft,  a-\-  ft,  I  —  x),  and  Q^  will  become  equal  to  Q.  We  must 
show  now  what  the  values  of  the  coefficients  of  0,  and  Q^  become 
under  this  hypothesis.     We  have 

.    _  r{a  +  ft-\-l-y) 


r{a)r{ft)r{a-\^i-  y)r{ft+  i  -  y) 
[r(i  -  y)r{a)r{ft)  +  riy  -  i)r(a  +  i  -  y)r{ft  +  i  -  y)-]<p^ 

ria  +  ft+l-y) 


r{a)r{ft) 


r{r)Q.. 


Let  y  tend  to  unity ;  the  coefficient  of  Q  reduces  to  —  -p;/   ^  L  l 

^  r{oc)r{fty 

the  first  factor  of  the  coefficient  of  0^  has  -j= — ^^- —       .^  for  limit ; 
the  second  factor  can  be  written  in  the  form 

r(2  -  y)r{a)r{ft)  -  r{y)r{a  + 1  -  y)r(ft  +  i  -  r) 

I  -y 

The  limit  of  this  will  be  found  by  taking  the  derivative  of  the  nu> 
merator  for  y  =  i,  and  is 

2r'{i)r{a)r{ft)  -  r{a)r{ft)  -  r{ft)r{a). 

We  have,  therefore, 

(21)     F{a,  ft,  a  -\-  ft,  I  —  x) 

- r{a)r{ft)L  ^  ^'^     r{ft)      r(^)J^^'^' ^' ^'^^     r{a)r{ft)^' 

Second  Case. — Let  us  suppose  y  greater  than  unity,  and  write 
y  =  2m  where  in  is  an  integer  and  may  be  zero.  Let  us  examine 
the  series  F{a  -{-  \  —  y,  ft  -\-  i  —  y,  2  —  y,  x)  when  y,  supposed  at 
first  to  differ  a  little  from  2  -\-  vi,  tends  towards  this  value. 


250  LINEAR  DIFFERENTIAL  EQUATIONS. 

The  first  terms  of  the  series  are 
(a  +  I  -  y){I^  +  I  -  r)  „ 


1  + 


I  •  (2  -  r) 


■^  I  .  2(2  -  y){i  -y)  -^  "  ' 


+ 


^  (or  +  I  —  y){a  -\r  2  —  y)  .  .^.  {a -^  m -\- l  —  y)X  ^ 

1.2. .  .(^;^+iX2-x)(3-r)  •  •  •  ('«+2-r) 


L;t;m+I_^    ..    . 


As  y  tends  to  the  value  2  -f-  w,  the  terms  in  x,  x"" ,  .  .  .  ,  x" 
preserve  finite  values,  while  the  coefficient  of  ;r'"  +  '  becomes  infinitely 
great  unless  some  factor  in  the  numerator  vanishes.  In  order  that 
this  may  happen  it  is  necessary  and  sufficient  that  a  or  ^  take  one 
of  the  values  i,  2,  .  .  .  ,  in  -{-  1.  In  this  case  all  of  the  other  terms 
will  retain  finite  values  for  y  =  2  -}-  in,  and  we  shall  still  have  an 
integral.  The  integral  is  uniform  in  the  region  of  the  origin,  and 
admits  this  point  as  a  pole  ;  the  integral  is  thus  seen  to  be  different 
from  F{a,  ft,  y,  x).  Starting  from  the  term  in  x'" ,  the  aggregate  of 
the  remaining  terms  may  manifestly  be  written 

CF{a,  /3,  m^2,x)x'"  +  '', 

as  to  the  preceding  terms,  they  may  be  replaced  by 

I 


C,F{^cx,  a-\-i  -y,  a-]ri-  ft,  ~jx'"  +  ^-% 

so  that  after  multiplying  by  ;ir"''"  +  '^  the  integral  becomes 

C,x-''F(o!,  a  -\-  1  —  y,  a  -^  I  —  ft,  -j  -\-  CF{a,  ft,  y,  x\ 

Take  for  example  the  differential  equation 

.dy         dy 

which  corresponds  to 

ar  =  I,     ft  =  —  2,     ^  =  2. 


GO  UK  SAT:  HYPERGEOMETRIC  SERIES.  2$  I 

This  equation  admits 

as  particular  integrals ;  the  general  integral  is  then 

It  is  to  be  remarked  that  when  the  circumstance  above  signalized' 
presents  itself,  the  origin  is  not  a  critical  point  for  the  general  in- 
tegral, but  may  be  a  pole.     Removing  this  special  case,  we  see  that. 


the  series 


F{a  -{-  I  —  y,  /3  -\-  I  —  y,  2  —  y,  x) 


presents  terms  which  increase  indefinitely  when  y  tends  to  the  value 
2  -\-  m. 

We  will  now  seek  the  limit  of  this  integral  when    y  tends  to 

2  -\-  m: 

{in  +  2  -  y)F{a  +  I  -  r.      /^  +  I  -  K'      2  —  ;k,      x) 
The  first  term  in  the  []  to  become  infinite  is 


(^^^  +  01  (2  —  r)l  (^^^  +  2  -  r) 


x"'^\ 


Letting  now  y  tend  to  the  value  2  A^  m,  the  terms  in  i,  x,  x^,  .  .  .  , 
x'"-  vanish,  and  the  remaining  terms  are  finite  ;  for  example,  the 
term  in  ;ir'"  +  ^  becomes 


{a  —  m  —  i)\{a  —  i)(/?—  m  —  i)|(/?  —  i) 

in-\-  \\  .  —7n\ 


X 


■m  -\- 1 


where  —  in\  =  (—  in){—  ni  +  0  •   •  •  (—  0-    The  series  tends  there- 
fore towards  the  value 

P      ^        ~ (-ir;;d.;;.+  i| F{a,  ^.  y,  x).. 


I 


■252  LINEAR  DIFFERENTIAL  EQUATIONS. 

If,  therefore,  we  consider  the  integral 

( —  i)'"  in  \.  VI  -\-  \Hm-\-2  —  y) 


{a—m—\)\{a-\){^li—vi—\)\{fi—\) 


x^-yJ^{a-\-i—y,  ^-{-i  —  y,  2  —  y,  x), 


we  see  that  for  y  t=  2  -{-  in  this  becomes  F{a,  /?,  y,  x).  As  before, 
a  new  integral  can  be  found  by  seeking  the  limit  for  y  =^  2  -\-  in  of 
the  expression 

F,  —  F{a,  ft,  y,  x) 
2  -\-  ni  —  y 
this  is 

1  r-r      a  \    1    dp  I   dp  I      dF 

m  which  y  =  2  -\-  in.  We  can  then  express  this  new  integral  in 
terms  of  the  already  known  integrals  0,  and  0,  by  the  same  process 
as  that  above  employed.  We  operate  in  exactly  the  same  manner 
ii  y  —  a  —  ft,  or  a  —  ft,  is  an  integer. 

18.  We  will  now  apply  the  gerusral  theory  to  the  case  of  the  equa- 
tion 

which  is  obtained  by  making  a  =  ft  =  i,  y  =  i.  This  equation 
presents  itself  in  the  theory  of  the  elliptic  functions  when  we  wish 
to  define  the  complete  integral  of  the  first  kind, 


/„     V{t-x')(l-k'x') 

as  a  function  of  the  modulus  for  imaginary  values  of  the  latter.  The 
equation  has  been  studied  by  Fuchs*  from  this  point  of  view,  and 
studied  directly  by  Tannery.f 

The  results  which  these  writers  have  obtained  are  derived  with- 
out difficulty  from  the  general  case.  For  this  example  we  will  em- 
ploy Tannery's  notation. 

*Crelle,  vol.  71,  p.  91. 

f  Annales  de  VAcole  normale  sup/rieure,  2*  serie,  t.  viii. 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


253^ 


Equation  (i)'  admits  an  integral,  P  say,  which  is  uniform 
throughout  the  plane,  provided  the  path  of  the  variable  never  crosses 
the  line  i 1-  co.     Inside  the  circle  C^  we  have 


P=F{\,  i   I,  x)=  1+^^ 


1.3.5...  i?^>i  -  0 


Let 


2.4.6 

cp{x)  =  F{h    h    I,    x). 
In  the  space  E^  we  shall  have,  in  like  manner, 

I 


2  m 


]■•■• 


P  = 


v\ 


0 


Since  y  =  i,  we  shall  have  a  new  integral  containing  a  logarithm.. 
Denote  this  integral  by  Q.  This  integral  will  be  uniform  throughout 
the  plane  if  the  path  of  the  variable  does  not  cut  either  of  the  lines 

—  00 o,  I 1-  00.     Write 


tp{x)  =  2 


{2111  —  \) 


L       2.4.. 


^3*5^  '    2111 


21  n      _ 
I 


I 


I 

2Vl 


then  inside  the  circle  C^  we  have 

Q  =  At{x)  +  0(,r)  log  X, 
and  in  the  space  E^ , 


Q  = 


=  j  ^  (j^)  [log  -  -  log  (I  -  x)\  +  4.A  (^)  }. 


^i- 


I 


Equation  (i)'  does  not  change  form  when  we  replace  x  by  \  —  x^ 
and  it  therefore  admits  two  other  integrals,  P'  and  Q' ,  which,  subject 
to  the  same  conditions  as  the  preceding  ones,  are  uniform  through- 
out the  plane.     In  the  circle  C^  we  shall  have 

P'  =  0(1  -x\ 

Q  =  0(1  -  x)  log  {i  -x)  +  4//<i  -  x)  ; 


i 


^54 


LINEAR  DIFFERENTIAL   EQUATIONS. 


in  the  space  E^ , 


P'  = 


Vx 


Q' 


I    ^  ,  /^  —  I 


Vx 


I  0  (^-)  [log  (I  -  ^-)  -  log  ^]  +  4^'  ('V^)  I  ^ 


These  four  integrals  suffice  to  express  any  integral  whatever  in 
the  entire  extent  of  the  plane.  In  order  to  find  the  linear  relations 
existing  between  these  integrals,  write  in  formula  (21)  a  =  |-,  /?  =  -I, 
y  ^  I  \  we  find  then 


71 


r(i)  —  — ^- 


7t 


or 


n  n 


also, 


We  have  now 


P 


4  log  2  _,       I    ^, 
5_  p'  —  -  Q', 

7t  7t^ 


(22) 


Q 


16  log'' 2  —  71^  4  log  2 


7t 


Q. 


7t  7t 


^  16  log^  2  -  n'  ^,  _  4  log  2  Q,^ 


These  are  the  relations  found  directly  by  Tannery.  These 
formulae  suffice  to  integrate  the  equation  (i)',  a  fact  which  has  been 
already  observed  in  the  general  case.  As  an  application,  we  will 
take  the  example  treated  in  Tannery's  memoir.  Suppose  we  start 
from  a  point  a  (Fig.  6)  very  near  the  point  x  =  2,  and  in  the  upper 


GOURSAT:  HYPERGEOMETRIC  SERIES. 


255 


lialf  of  the  plane,  and  describe  a  closed  path  including  in  its  interior 
the  points  o  and  i. 


Fig.  6. 


This  path  can  be  reduced  to  a  path  aMA,  where  y^  is  a  point 
of  the  line  o i,  say  the  point  ^  =  |-,  followed  by  two  loops  de- 
scribed successively  in  the  direct  sense  round  the  points  x  ^  o  and 
jr  =  I,  and  then  the  path  A  Ma.  Let  us  start  from  the  point  a  with 
the  integral  P' ;  we  will  of  course  arrive  at  the  point  A  with  the 
same  integral.  Now  to  find  the  change  in  P'  when  we  go  round  the 
loop  X  :=  o,  we  replace  P'  by  its  value, 

Tt  7t 

After  describing  the  loop  O,  /'returns  to  its  original  value,  but   Q 
changes  into  Q  -\-  2niP^  and  therefore  P'  becomes 

P'  -  2iP. 

Now  to  see  how  this  integral  behaves  on  going  round  the  loop  i,  we 
replace  P  by 

I 


_      4  log  2  „, 

p  ^         o         p' 


Q'\ 


P'  will  not  change,  but  Q  becomes  Q  -\-  27tiP',  so  that  P'  —  2iP 
changes  into 

-  3P'  -  2iP. 

We  return  to  a,  therefore,  with  the  integral 

xn  -\-^i  log-  2  _,       2i  ^, 

-  ^-^ ^-  P'  +  —  Q. 

n  7t 


256  LINEAR   DIFFERENTIAL   EQUATIONS. 

If  we  start  from  a  with  Q' ,  we  arrive  at  A  with 
16  log'  2  —  TT^  ^      4  log  2 

7t  7t  "^ 

After  the  loop  o  the  integral  will  be  represented  by 

16  log''  2  —  tt"  4  log  2  _ 

^^ P  -  ^*— ^—  Q  -  8z  log  2  .  P; 

or,  what  comes  to  the  same  thing,  by 

Q  -U\og2.P; 


or,  again,  by 


n  +  8?  log  2  _  32?  log'  2  p, 


After  the  loop  i  we  shall  have 

7t  4-  U  log  2       2/7r  (;r  +  8?  log  2)  -  32/  log'  2  ^  ^ 


that  is, 


2?(7r  +  4/log2)'  ^,       7r+8nog2 


I 


We  will  of  course  return  to  a  with  this  same  integral. 

The  equation  (i)'  admits  also  two  other  integrals,  susceptible  of 
development  in  series,  analogous  to  the  preceding  ones.  First  we 
have  the  integral 


\/  —  X      \xj        Vi—x     \i  —  X. 

which  is  uniform  throughout  the  plane  provided  the  path  of  the 

variable  does  not  cut  the  line  o 1- 00.     Denote  this  integral 

by  P'.  To  find  another  integral,  suppose  first  y=  i,  o' =  4-,  and 
let  /:f  differ  a  little  from  the  value  ^.  The  differential  equation  will 
admit  the  two  integrals 


GOURSAT:   HYPERGEOMETRIC  SERIES. 
The  expression 


257 


v 


=4.pf-^.i)-(-)-'HAA/'+j.i) 


ft 


will  also  be  an  integral.     The  limit  of  this  expression  when  /?  =  •!■ 
will  be  a  new  integral  Q' ,  viz., 


e"=-iog(-.),(i)-^^  +  ^-,(i) 


also, 


\  \  —  X 


-**(r^)'°gc— )+#(y^)]. 


Q"   is  uniform   throughout  the    plane  under  the    same    conditions 
as  P". 

For  all  values  of  x  in  the  upper  half  of  the  plane  we  have  the 
relations 


(23) 


\ 


4.1o8r  2  I 

n  71  -^   ^ 

,   _    7t   -  \i  log   2  ^,  i    ^„_ 


P 


P'  = 


for  values  of  .r  in  the  lower  half  of  the  plane  we  have 

[ 

I 

(24)  \ 


{  4loof  2  I 


I   P'  =  ^  +  4^'  log  2^„  _ 


Q". 


These  formulae  are  established,  like  (21),  by  starting  with  the  rela- 
tions (13)  and  (17)  which  exist  in  the  general  case. 

Remark. — Formulne  (23)  and  (24)  do  not  appear  to  be  in  accord 
with  the  formulse  given  by  Tannery  {loc.  cit.  p.  188).  This  arises 
from  the  fact  that  the  functions  P'  and  Q"  here  used  are  not  pre- 
cisely the  same  as  the  corresponding  ones  employed  by  Tannery. 


258  LINEAR  DIFFERENTIAL  EQUATIONS. 

Instead  of  the  system  of  integrals  P"  and  Q'  consider  the  following 
system: 


P."  = 


where  the  argument  of  x  is  supposed  to  lie  between  —  n  and  -|-  it. 
We  find  easily  for  the  upper  half  of  the  plane 

p"  =  ip;',     Q"  =  iq:'  -  TtPr-, 

and  substituting  these  values  in  the  second  of  (23),  we  have 

■'  A  loP"  2  I 

which  is  identical  with  Tannery's  relation. 

19.  The  formulae  established  in  what  precedes  enable  us  to  deter- 
mine whether  or  not  the  differential  equation  admits  an  algebraic 
integral  and  whether  its  general  integral  is  or  is  not  algebraic.  This 
question  has  been  treated  by  Schwarz  (Crelle,  t.  73,  p.  292),  who,  by 
employing  the  differential  equation  of  the  third  order  satisfied  by 
the  ratio  of  two  particular  integrals  of  equation  (i)  and  by  the  aid 
of  Riemann's  surfaces,  was  led  to  a  question  in  spherical  geometry 
where  regular  polyhedra  presented  themselves.  Later,  Klein  con- 
sidered the  more  general  case  of  a  linear  differential  equation  of  the 
second  order  with  rational  coefficients.  In  the  following,  Schwarz"s 
results  are  arrived  at  by  quite  elementary  considerations. 

If  equation  (i)  possesses  a  single  algebraic  integral,  this  integral 
must  reproduce  itself  to  a  constant  factor  pres,  when  we  turn 
round  a  critical  point.  In  the  domain  of  the  point  ;i'  ^  o  it  will  be 
represented  by  one  of  the  integrals  0, ,  0^ ,  and  in  the  domain  of 
;ir  =  I  it  will  coincide,  to  a  factor  pres,  with  either  0^  or  06-  Refer- 
ring now  to  the  relations  existing  among  these  four  integrals,  we 
see  that  we  cannot  have  an  integral  of  the  required  kind  unless  one 
of  the  numbers  a,  f3,  y  —  a,  y  —  ft  is  an  integer.     In  each  of  these 


GOURSAT:  HYPERGEOMETRIC  SERIES.  259 

cases  one  of  the  hypergeometric  series  which  express  the  general 
integral  will  have  a  limited  number  of  terms.  Suppose  for  example 
j/  —  a  =  —  m  ;  equation  (i)  will  then  have  the  integral 

(i  -  x)y-''-^P, 

P  being  an  integral  function  of  ;r.  In  order  that  this  integral  shall  be 
algebraic,  it  is  further  necessary  that  /3  be  a  real  and  rational  num- 
ber. If  equation  (i)  admits  more  than  one  algebraic  integral,  the 
general  integral  will  be  algebraic.  We  propose  now  to  find  all  the 
cases  where  this  is  true.  We  can  exclude  from  our  research  the 
cases  where  one  of  the  numbers  y^  y  —  a  —  ;S  is  an  integer ;  in  fact, 
we  have  seen  that  in  these  cases  there  exists  a  logarithm  in  the 
complete  integral  in  the  region  of  one  of  the  critical  points  ;  as  an 
exceptional  case  this  logarithm  may  disappear,  but  then  the  point 
considered  is  no  longer  a  branch-point  for  the  general  integral,  and 
it  will  sufifice  to  examine  how  an  integral  behaves  in  the  region  of 
the  other  critical  point  in  order  to  be  certain  of  its  nature.  We  see 
further  that  the  three  numbers  a,  /?,  y  must  be  real  and  rational;  if 
this  were  not  so,  then  in  the  domain  of  the  critical  points  there 
would  exist  an  integral  which  could  take  an  infinite  number  of 
values.  This  being  granted,  suppose  now  that  y^  and  y^  are  two 
linearly  independent  particular  integrals,  uniform  throughout  the 
plane  provided  the  path  of  the  variable  does  not  cross  either  of  the 

lines  —  c» o,  i 1-  00  .     Start  from  an  arbitrary  point  A 

of  the  plane  with  an  integral  cy^  -|-  c'y^  and  describe  an  arbitrary 
■closed  path  which  does  not  pass  through  either  of  the  points  ;f  =  o 
or  ;ir  =  I.  We  return  to  the  point  of  departure  with  an  integral 
Cy^  -\-  C  y^.  If  this  integral  is  an  algebraic  one  the  coefficients 
C  and  C  admit  only  a  limited  number  of  values  whatever  be  the 
path  described.  If  this  is  true,  then,  whatever  be  the  original  coef- 
ficients c  and  c\  the  general  integral  is  algebraic.  Consider  in  fact 
the  logarithmic  derivative  of  the  preceding  integral,  viz., 

C 

cy:+c'y:  ^  ^''  +  ~c  ^^ 

Cy.  +  C'y^  C       ■ 


26o  LINEAR   DIFFERENTIAL   EQUATIONS. 

The  ratio  —^  admitting  only  a  limited  number  of  values,  the  same  is 

true  of  this  logarithmic  derivative,  and  as  besides  it  admits  of  no 
other  singular  points  than  critical  points  and  poles,  it  follows  that  it 
is  an  algebraic  function  of  x. 

This  being  true  for  any  integral,  let  -s,  and  s^  denote  two  inte- 
grals of  (i);  we  have  then  the  relation 


-,5'/  =  Ha--'>'(i  —  x)i-'^-^-\ 
which  can  also  be  written 


z,z„  — 


The  product  of  any  two  integrals,  z^ ,  s^,  is  then  an  algebraic  func- 
tion of  X.  If  z^  and  z^  are  two  integrals,  then  z^  and  z^  -\-  z^  will  also- 
be  two  integrals.  The  product  z^^  -\-z^z^,  and  consequently  z^,  will 
be  an  algebraic  function  of  the  variable. 

Every  closed  path  starting  from  and  returning  to  a  point  A  can 
be  reduced  to  a  series  of  loops  described  round  the  points  ;f  =  o 
and  X  =  I  ',  we  are  therefore  led  to  examine  the  behavior  of  the 
preceding  ratio  when  we  describe  one  of  these  loops.  We  will  take 
for  J,  and  jj  the  integrals  0,  and  0^ ;  then  starting  from  a  point  A 
with  the  integral  (70,  -|-  C (p^  and  describing  the  loop  ;ir  =  o  in  the 
direct  sense,  we  shall,  as  we  have  seen  above,  return  to  A  with  the 
integral  C^cp^  +  C^'cp^ ,  where 


r  -  r  4-  /-/-^(^  +  /^  + 1  -  r)-^(i  -  r)  (.  _  .../d-v^ 


I 


C/=  C'e^'^'^'-y). 


GOURSAT :   HYPERGEOMETRIC  SERIES.  261 

c 

Let  p  be  the  initial  value  of  -^,  and  p'  its  final  value ;  then 


0'=  ^.— -u-v)  +  r{a  +  ft+i-  y)r{i  -  r)fi  -  .--'C-vA 

ing 


this  last  formula  becomes 

{A)  p'  -  a  =  K{p  -  a). 

If  the  loop  were  described  in  the  negative  sense  we  should  have 

{Ay  p'  -  a  =  ^{p-  a). 

Now  let  the  variable  describe  the  positive  loop  around  ;r  =  i  ;  the 
integral  C<f>^-{-  C'tp-i  changes  into  (7,0,  +  ^/03>  where 

r{y  —  a)[{y  —  fty 

Denoting  again  by  p  and  p'  the  initial  and  final  values  of  the  ratio 
-,  we  have 


C 


p'        p  ^  r(v  —  a)r(v  —  /3)\     ^^«(v-a-P)    J 


p         p  '    r  {y  —  a)r  {y  —  0)\     ^^^(v-a- 

Making 

r{y)r{y-a-(3)  _  _ 
r{y-a)r{y-^)-         ^' 

^-2Jr/(y-o— P)    ^    J<^' 


262  LINEAR  DIFFERENTIAL  EQUATIONS. 

we  have,  from  the  last  equation, 

if  the  loop  is  described  in  the  negative  sense,  we  have 


{B)' 


'-,-b 


h%'^^- 


The  problem  which  we  seek  to  solve  can  be  stated  as  follows : 
Having  given  a  scries  of  quantities  such  that  each  is  deduced  from 
the  preceding  by  one  of  the  formulce  {A),  {A)',  {B),  {B)',  in  what  cases: 
can  we  arrive  at  a  limited  number  of  different  quantities,  the  order  in 
which  these  fornudce  are  successively  applied  being  perfectly  arbi- 
trary ? 

The  geometrical  method  seems  to  conduct  most  readily  to  the 
sought  result. 

(20)  Draw  in  the  plane  two  rectangular  axes  O^and  Orj.  Represent 
as  usual  the  quantity,/?  =  ^  -|-  irj  by  the  point  whose  co-ordinates 
are  $  and  ri\  let  ^  and  B  denote  the  points  (on  the  axis  O^)  which 
represent    the    quantities    a   and    b   respectively.       Let    M  be    the 

point  representing  the  initial  value 
of  p,  and  M'  be  the  point  represent- 
ing the  value  of  p'  obtained  by  letting 
the  variable  travel  round  the  loop 
;r  =  o  in  the  direct  sense : 


p   —  a 


K{p-a). 


The  quantity  p  —  a  will  be  repre- 
sented by  the  extremity  of  a  segment, 
drawn  through  the  origin,  equal  and 
parallel  to  the  segment  AM\  if  this 
the  point  A  through  an  angle  gi? 
(&?  =  27T{y  —  i)),  we  obtain  the  segment  AM',  of  which  the  ex- 
tremity M'  represents  the  quantity  p.  If  the  loop  were  described 
in  the  negative  sense,  we  should  come  to  the  point  J//  obtained,  as 
before,  by  making  the  segment  yi  J/ turn  round  the  point  A  through 
an    angle   00,   the   turning,   however,   being   in   the  negative    sense 


Fig.  7. 

segment    is    turned    round 


GO  URSA  T:  HYPERGEOAIETRIC  SERIES. 


263 


When  the  variable  describes-a  series  of  loops  round  O,  the  different 
values  of  p  will  be  represented  by  the  vertices  of  a  regular  polygon 
inscribed  in  the  circle  of  radius  AM,  one  of  these  vertices  being  the 
point  M. 

Suppose  now  (Fig.  8)  that  the  variable  describes  a  loop  round 
the  critical  point  x  =■  \.     Let  M  be  the  point  denoting  the  initial 

value  of  p ;  -  will  be  represented   by  the 


point  M^, ^   by  the  extremity  of  a 

segment  drawn  through  the  origin  equal 

and  parallel  to  BM, ,  and  iT'  f-  -  b\  by 

the  extremity  of  a  segment  equal  and 
parallel  to  the  segment  BAI^' ,  which  is 
simply  the  segment  BAI^  turned  round  B 
through  an  angle  00'  =  2n{a.  -(-  /3  —  y^. 


Fig.  8 


Consequently  the    point  J//  represents    the    quantity  —7 ,  and  the 
point  M'  represents    the    quantity   p'.     Let   the  variable  describe 


several  successive  loops  round  the  point  x 


I  ;  the  point  —  will 
P 


then  coincide  successively  with  vertices  of  a  regular  polygon  in- 
scribed in  the  circle  whose  centre  is  at  ^  ;  the  point  p  itself  will  be 
on  the  circumference  of  a  circle  transformed  from  the  preceding  by 
reciprocal  radii  vectores  having  the  origin  for  the  pole  of  the  trans- 
formation, and  unity  for  the  modulus.  If  we  apply  this  transforma- 
tion to  all  of  the  circles  having  B  as  the  centre,  we  evidently  obtain 
a  system  of  circles  passing  through  two  fixed  imaginary  points.  It 
is  easy  to  see  what  are  the  point-circles  of  this  system':  first  is  the 
origin,  which  corresponds  to  the  circle  of  infinite  radius  with  centre 


at  B,  and  then  the  point  C, 


-, -corresponding  to  the   circle  of 


radius  zero.  All  the  circles  of  this  system  are  conjugate  with  respect 
to  these  two  points,  which  are  two  double  points  of  the  homographic 
transformation 


.  =  ^'--. 


264 


LINEAR  DIFFERENTIAL  EQUATIONS. 


Before  going  farther  we  shall  demonstrate  the  following  geometrical 
theorem  : 

Having  given  a  circumference  and  n  points,  A,  B,  C,  .  .  .  ,  L,  upon 
this  circumference  which  are  so  disposed  that  on  making  an  inversion, 
taking  for  the  pole  a  point  O  of  the  plane,  the  corresponding  points 
A',  B',  .  .  .  ,  L'  are  the  vertices  of  a  regular  polygon,  if  O'  is  the  con- 
jugate point  to  O  with  respect  to  the  circumference,  all  the  points  of 
the  circumference  described  upon  00'  as  a  diameter  and  perpendicular 
to  the  plane  of  the  figure  will  possess  the  same  property  as  the  point  O. 


Let  5  be  a  point  of  this  circumference ;  then,  from  an  elementary 
property  of  the  inversion, 

AB 


A'B' 

~  OA  .  OB    ' 

A,B, 

-      ^^      P- 
~  SA  .  SB^'' 

AA 

A'B'  ~ 

fPA  _  OA  ^^  OB 
[pj^SA^SB 

consequently 


.       OA     OB  .  J  t.  •    .  r 

Each  of  the  ratios  -^  ,  -^  is  constant,  and  so  the  same  is  true  of     -] 

A  B  ' 

the  ratio  -r— .     If  the  polygon  A'B'C'D'E'F'  has  its  sides  equal,     3 

A  B 

the  same  will  be  true  of  the  polygon  A^B^C^Dfi^x  •     Q-  E.  D. 


GOURSAT:   HYPERGEOMETKIC  SERIES. 


265 


Fig.  10. 


Returning  now  to  the  proposed  question,  consider  the  circum- 
ference described  upon  OC  (Fig.  10)  as  diameter  in  a  plane  perpen- 
dicular to  the  plane  of  the  figure.     Let  5  and  S'  be  the  points  of 
intersection  of  this  circumference  with  the 
perpendicular  to  the  plane  of  the  figure 
through  the  point  A.     Conceive  a  sphere 
described  upon  55' as  diameter,  and  make 
a  projection  upon  this  sphere,  taking  5  as 
the  point  of  sight. 

All  circles  having  their  centres  at  A 
are  projected  upon  circles  having  for  poles 
the  points  5  and  S'.  As  to  circles  con- 
jugate with  respect  to  two  points  O  and  C,  they  are  projected  upon 
circles  having  PP'  for  axis.  It  follows  from  what  precedes  that 
having  given  upon  the  surface  of  the  sphere  a  point  m  representing 
the  value  of  p,  we  shall  find  the  point  ;//  representing  a  new  value 
of  p  by  turning  the  point  m  through  an  angle  oo  round  SS'  or 
through  an  angle  go'  round  PP'.  This  is  evident  for  the  axis  SS'; 
as  to  the  axis  PP'  it  sufifices  to  remark  that,  having  given  two  points 
M,  M'  in  the  plane  representing  two  values  of  p,  of  which  one  is 
deduced  from  the  other  by  one  of  the  formulae  {B\  (B)',  if  we  make 
an  inversion  with  the  point  O  for  pole,  the  angle  M^BM/  is  equal 
to  go',  and,  from  the  preceding  theorem,  the  angle  jnpm'  must  have 
the  same  value,  p  denoting  the  foot  of  the  perpendicular  let  fall 
from  m  upon  the  axis  PP'. 

We  can  now  replace  the  above  enunciation  of  our  problem  by 
the  following : 

Having  given  tivo  diameters  SS'  and  PP'  in  a  sphere  and  a  series 
of  points  upon  the  surf  ace  zvhich  succeed  one  another  by  a  law  such  that 
we  pass  from  any  one  to  the  following  one  by  turning  through  an 
angle  go  round  SS'  or  an  angle  go'  round  PP' ,  in  what  cases  can  we 
arrive  at  a  limited  number  of  points,  the  order  in  which  the  construc- 
iiotis  are  applied  being  perfectly  arbitrary  ? 

All  depends,  evidently,  upon  the  order  of  symmetry  of  the  axes 
SS'  and  PP'  and  on  the  angle,  V,  between  them.     Let 


/  P' 


266  LINEAR  DIFFERENTIAL   EQUATIONS. 

P  P'  •  • 

The  fractions-  and  —.  beinj;^  irreducible,  SS'  will  be  an  axis  of  sym- 

q  <1 

metry  of  order  q,  and  PP'  one  of  order  q' .     As  to  the  angle  V,  we 
have  from  the  preceding  figure 

F=  2AOS, 

^_„       OA  OA  I'OA 

cos    A  OS    =     -7:^r 


OS         VOAxOC       V  OC 

cos  V  =  '2.Y)r ^  ~  -^'^  ~  ■^* 

Replacing  a  and  b  by  their  values,  we  get 

^^'  ^  -  T{a  +  I  -  r)^(/^+  I  -  r)  ^  r(;/  -  a)/Xr  -  ft)      ' 

or 

sin  iy  —  oi)Tt  sin  (v  —  /S);r 

cos   K  =  2—^ : — ; ^7 — ■  —   I. 

sni  yit  Sin  yy  —  a  —  p)7i 

Remark. — When  we  diminish  or  increase  the  value  of  one  of  the 
quantities  a,  fi,  y  by  any  number  of  units,  q  and  q'  do  not  change, 
neither  does  cos  V.  We  may  then  suppose  i  —  y  and  y  —  a  —  fi 
to  lie  between  o  and  i,  and  can  thus  conduct  a  —  y^  to  lie  between 
—  I  and  -|-  I  ;  but  as  there  is  symmetry  between  the  elements. 
a  and  /3,  we  may  suppose  a  —  ft  \.o  lie  between  o  and  i,  a  supposi- 
tion which  will  be  adopted  in  what  follows. 

(21)  The  geometrical  representation  upon  the  sphere  is  not  possible 
unless  the  point  A  lies  between  the  points  O  and  C,  or,  what  comes 
to  the  same  thing,  unless  the  product 

sin  (;/  —  a)TT  sin  {y  —  ft)7r 
"~  sin  yn  sin  {y  —  a  —  ft)7r 

is  comprised  between  o  and  i .  In  the  contrary  case  we  can  demon- 
strate directly  that  the  ratio  p  can  take  an  infinite  number  of  values 
in  each  point  of  the  plane.  The  establishment  of  this  last  state- 
ment is  based  upon  the  following  remarks  : 

I.  There  exists  a  circle  having  its  centre  at  A,  and  conjugate 
with  respect  to  the  segment  OC.     If  the  point  J/ which  represents 


1 


CO  URSA  T:  HYPERGEOMETRIC  SERIES.  267- 

the  initial  value  of  p  is  on  this  circle,  all  the  other  points  derived 
from  it  will  likewise  be  on  the  circle.  If  the  point  M  \?,  outside  or 
inside  the  circle,  the  derived  points  will  themselves  be  all  outside  or 
all  inside. 

II.  Supposing  that  we  have  taken  the  point  A  as  origin,  let  us 
denote  by  z  and  z'  the  quantities  which  are  represented  by  the 
points  M  and  M'  in  the  new  system. 


Fig.  II. 

The  first  transformation  is 

z'  =  kz', 
in  the  second  transformation  z'  is  connected  with  ^  by  a  relation 

cz^d' 
where 

a  =  a  ^  i^,  c  =  a'  -\-  i/S', 

b=yJ^id,  d=y'-\-i6'. 

Let  us  seek  the  locus  of  the  points  z  such  that  mod  z  =  mod  z';  we 
have 

J     ,        mod  (az  4-  b) 

mod  ^    = — i --—{  , 

mod  {cz  -f-  a) 

az  -{-  b  =  {ax  -  /3y  -^  y)  ^  i{(3x  -\-  ay  -\-  d), 


mod  {az  -\-b)  =  V{a'  +  f:f){x'  +  y')  +  mx  +  ny  +/, 


mod  {cz  +  d)  =  V{a"  +  I3"){x'  +/)  -f-  m'x  +  n'y  +/'. 
The  equation  of  the  sought  locus  is 

{a'  +  ff){x'  +/)  +  mx  -^ny  -\-p 

=  {a''  +  /5-)(^=  +  yj  +  {m'x  +  n'y  +p'){x^  +/).. 


268 


LINEAR   DIFFERENTIAL   EQUATIONS. 


This  is  a  quartic  curve  having  the  circular  points  at  infinity  for 
double  points.  The  circle  with  centre  A  (Fig.  ii),  conjugate  to  the 
segment  OC,  is  evidently  part  of  the  locus,  and  as  the  points  O  and 
C  also  belong  to  it  the  quartic  breaks  up  into  two  circles,  viz.,  the 
circle  {A)  and  another  circle  through  the  two  points  O  and  C. 

If  the  point  M,  which  represents  the  value  of  z,  is  on  the  ex- 
terior or  the  interior  of  the  two  circles,  we  have 


M,     M 


\ 


mod  z'  <  mod  z ; 
but  if  z'  is  outside  one  circle  and  inside  the  other,  we  have 
mod  z'  >  mod  z. 

This  granted,  let  J^be  the  point  representing  the  inital  value  of  p, 

and  suppose  M  outside  the  two  circles  {A),  (^)'- 

Applying  to  the  point  Tlf  the  second  transformation  as  often  as  it 

gives  us  different  points,  we  shall  arrive  at  a  certain  number  of  points 

upon  a  circle  conjugate  to  the  segment 
OC.  Suppose  tJ/j  that  one  of  these 
last  points  which  is  nearest  A  ;  M^ 
must  therefore  lie  inside  the  circle  A' , 
and  we  have  AM^<  AM.  Apply 
now  the  first  transformation  to  the 
point  M^  so  as  to  obtain  a  point  M^ 
outside  the  circle  A' ;  we  shall  have 
AM^  =  AM, .  Apply  the  first  trans- 
formation to  the  point  M^ ;  we  shall 
find  as  before  a  point  M^  nearer  A 
than  the  point  Af^ ,  etc.  We  shall 
find  by  continuing  this  process  a 
series  of  points  M,  M, ,  M,,  AI,,  .  .  . 

following  one  another  according  to  such  a  law  that  we  shall  never 

have  a  new  point  coinciding  with  one  of  the  preceding  ones,  and 

this  will  go  on  indefinitely.     The  ratio  p  takes  therefore  an  infinite 

number  of  values. 

(22)  The  question  in  spherical  geometry  to  which  we  have  been 
led,  which,  as  we  shall  presently  see,  is  identical  with  the  question  to 


Fig.  12. 


GO  URSA  T :   HYPERGEOMETRIC  SERIES.  269- 

which  Schwarz  was  led,  has  been  solved  by  Steiner.*    We  can  also  de- 
duce the  solution  from  a  memoir  by  Jordanf  upon  Eulerian  polyhedra. 
There  are  two  particular  cases  where  the  solution  is  immediately 
perceived. 

1.  If  we  have  two  axes  of  binary  symmetry,  the  angle  between 
them  must  be  commensurable  with  27r. 

2.  If  we  have  one  axis  of  binary  symmetry,  and  one  axis  of  order 
of  symmetry  =  TT perpendicular  to  the  first,  we  shall  always  end  with 
a  limited  number  of  points. 

Discarding  these  particular  cases,  let  us  suppose  that  we  have 
two  axes   of  symmetry  PP\   SS',  one   of  which,  PP',  is  of  order 

K{K  _  3).     The  repetitions  by  symmetry  of  the  axis  PP'  will  be 

axes  of  symmetry  of  order  K  of  the  figure  formed  by  the  symmetric 
repetitions  of  a  point  JM  of  the  sphere.  These  repetitions  ought  to 
be  limited  in  number.  This  comes  to  studying  the  figure  formed  by 
taking  P  itself  as  the  point  of  departure.  We  can  demonstrate 
without  difficulty  that,  if  we  arrive  at  a  limited  number  of  points, 
these  points  are  the  vertices  of  a  regular  polyhedron.  The  axes 
PP\  SS'  ought  then  to  be  the  axes  of  symmetry  of  a  regular  poly- 
hedron having  one  of  its  vertices  at  P.  It  is  evident  that  the  plane 
of  the  two  axes  will  be  a  plane  of  symmetry  for  the  polyhedron. 
This  necessary  condition  is  also  suflficient.  Conceive  in  fact  this, 
regular  polyhedron,  and  consider  the  spherical  polygons  on  the  cir- 
cumscribed sphere  which  correspond  to  its  different  faces.  The 
repetitions  by  symmetry  of  one  of  these  polygons  will  always  lead  to 
an  analogous  polygon  ;  as,  further,  one  of  these  polygons  can  only 
coincide  with  itself  in  a  limited  number  of  ways,  it  follows  that  the 
symmetrical  representations  of  a  point  will  be  limited  in  number. 

Example  I. — Let  ;/  =  !-,  ar  -|-  /?  =  O.  We  have  two  axes  of  binary 
symmetry, 

cos  F=  2  cos  na  cos  nfi  —  i  =  cos  2a7t. 

It  is  sufificient  that  a  be  commensurable.  This  can  be  verified 
directly.     The  differential  equation  is 

,  <^V    ,    /i  \dy 

*  Crelle,  vol.  xviii.  p.  295.  -f-Ibid.  vol.  Ixvi.  p.  22. 


270 


or 


LINEAR  DIFFERENTIAL   EQUATIONS. 


,  .d^ydy    ,    .  ^  (dyV  ,    dy 


From  this  we  have 
d 


x{i-x)i^-£f\^a^d{^-f\ 
dy 


adx 


Vc'  —  y'^        Vx{i  —  x) 


y  _      .— 

sin"'  -  =  cY  sin  '  i  x  A-  Ji. 
c 


If  a-  is  commensurable,  we  deduce  from  this  an  algebraic  relation 
between  x  and  y. 

Example  II. — Schwarz's  example  : 


r  =  f '     «'  =  —  tV  ' 


fi 


1 


y  -  (y  =  i,     r  —  /^  =  tV'     y  —  a  —  ft  =1. 

There  is  one  axis  of  binary  and  one  of  ternary  symmetry ;   their 
angle  is  given  by 


.    3^    .     S7t 
2  sm  —  sm  — 

COS  y  = I 


27r 


sm 


I 


This  is  exactly  the  angle  which  in  a  regular  tetrahedron  is  in- 
cluded between  the  altitude  and  the  line  joining  the  middle  points 
of  two  opposite  edges :  the  integral  is  then  algebraic. 

(23)  The  preceding  results  can  be  placed  in  a  little  different  form 
and    so    bring  better  into  evidence    their  identity  with   Schwarz's   | 
results. 

Suppose,  as  explained  above,  that  we  have  conducted  the  three 
numbers  i—  y,  y  —  a  —  /3,  a  —  /3to  values  lying  between  o  and  i. 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


2/1 


and  denote  by  A,  yu,  v  the  three  positive  numbers,  less  than  unity, 
so  obtained : 

A  E  I  —  X, 

}x~y  —  a  —  ^, 

v  =  a  —  /?. 

We  will  now  find  out  for  what  sys- 
tems of  values  of  these  numbers  the 
integrals  are  algebraic.  As  before, 
let  SS'  and  PP'  (Fig.  13)  be  the  two 
axes  of  symmetry. 

Construct  a  triangle  having  for 
base  SP,  and  for  angles  P^^^  [i—y)7r, 
SPQ  =  {y  -a-  f5)7r. 

For  the  angle  Q  we  have 


Fig.  13. 


•COS  Q  =  sin  P  sin  5  cos  {SP)  —  cos  P  cos  5 


(i)7r 


.     f  ^-     r^  sin  (v  —  a)7r  sin  (y 

=  sm  v;rsin  (y  —  a—/3)7t  1  — -. — — '- ^ 

L    sm  yTT  sin  (y  —  a  — /J)7r 

or,  finally,  -f-  cos  yrr  cos  (y  —  a  —  fi)7t, 

cos  Q  =  cos  {a  —  ft)7T. 

The  angle  Q  is  comprised  between  o  and 
supposed  a  —  fi  comprised  between  o  and  i 

.-.     Q  =  {a-  I3)7r. 


TT,  and  further  we  have 


The  three  angles  of  SPQ  are  then 

S  =  X7r,     P  =  ^7t,     Q  =  v7t. 

What  conditions  should  this  triangle  satisfy?  In  the  particular 
case  where  y=^,a-\-ft  =  o,  5  and  Z'  are  right  angles ;  the  point 
Q  is  then  one  of  the  poles  of  the  great  circle  SPS'P'.  If  we  con- 
struct a  double  pyramid  having  for  vertices  the  poles  Q  and  Q' ,  and 
for  base  the  regular  polygon  of  which  S  and  P  are  two  vertices,  the 
three  planes  of  the  trihedron  SPQ  will  be  three  planes  of  symmetry 
for  this  double  pyramid.  If  the  axis  SS'  is  a  binary  axis,  and  if  the 
arc  SP  is  a  quadrant,  the  three  planes  of  the  trihedron  will  still  be 
three  planes  of  symmetry  for  a  double  pyramid  whose  vertices  are 
at  the  points  P  and  P',  and  whose  base  is  in  the  plane  SQS'Q'. 


272  LINEAR  DIFFERENTIAL  EQUATIONS. 

These  singular  cases  being  examined,  let  us  see  what  takes  place 
in  the  general  case,  supposing  there  exists  a  regular  polyhedron  hav- 
ing SS'  and  PP'  as  axes  of  symmetry  and  having  one  vertex  at  P. 
The  three  planes  OSP,  OPQ,  OQS  will  be  three  planes  of  symmetry 
for  this  polyhedron  ;  it  is  evident  from  what  precedes  that  OSP  is  a 
plane  of  symmetry.  The  point  P",  symmetrical  to  P  with  respect 
to  the  plane  OSQ,  is  one  of  the  vertices  of  the  polyhedron ;  as  we 
can  start  from  F'  instead  of  P  to  find  the  remaining  vertices,  it  is. 
clear  that  the  plane  OSQ  is  a  plane  of  symmetry.  So  for  OQP;  viz., 
the  point  S"  is  an  extremity  of  an  axis  of  symmetry  of  the  same 
order  as  the  axis  SS' ;  if  we  replace  the  axis  OS  by  the  axis  OS",  we 
shall  obviously  form  the  same  figure,  and  consequently  OQP  is  a 
plane  of  symmetry.  , 

Thus  the  three  faces  of  the  trihedron  OPQS  are  three  planes  of 
symmetry  of  a  regular  polyhedron.  The  converse  is  easily  demon- 
strated ;  in  fact,  every  body  admitting  these  three  planes  of  symme- 
try OPS,  OSQ,  OQP  will  also  admit  OSP"  as  a  plane  of  symmetry. 
If  we  take  the  body  which  with  respect  to  the  plane  OSQ  is  sym- 
metric to  the  first  body,  then  take  the  body  symmetric  with  respect 
to  the  plane  OSP'  of  this  new  body,  this  last  will  coincide  with  the 
first  ;  but  these  two  operations  are  equivalent  to  turning  the  body 
through  an  angle  PSP"  =  2(1  —  y)7t  round  OS;  .-.  OS  is  an  axis 
of  symmetry  for  the  body.  Finally,  then,  in  order  that  the  general 
integral  shall  be  algebraic,  it  is  necessary  and  sufficient  that  the  three 
planes  of  the  trihedron  OSPQ  shall  be  the  three  planes  of  symmetry 
of  a  double  pyramid  or  of  a  regular  polyhedron.  It  is  necessary  first 
that  the  three  angles  Xn,  /att,  vtt  shall  be  the  angles  of  a  spherical 
triangle ;  this  requires 

A  +  ;/  +  7'  >  I, 

A  +  I  >  //  +  1^, 

^  +  I  >  I'  +  A, 

7'  +  I  >  A  +  /,/. 

These  conditions  are  equivalent  to  the  above-found  condition  that 

sin  {y  —  a)7r  sin  (;/  —  fi)7r 
sin  yTT  sin  (;/  —  a  —  f5)7r 

shall  lie  between  o  and  ,1. 


GOURSAT:  HYPERGEOMETRIC  SERIES. 


273 


It  evidently  comes  to  the  same  thing  whether  we  consider  the 
triangle  PQS  or  one  of  the  triangles  PQS\  QP'S',  QSP' ;  the  angles 
of  these  triangles  have  the  following  values : 

PQS AtT  fl7t  VTt, 

PQS' Att        (i  — //)7r     (i— y);r, 

P'QS (i  —  X)7t  juTt     (i  —  r)7r, 

P'QS' (i— A)7r       (i— /<)7r  vn. 

We  will  choose  the  triangle  for  which  the  sum  of  the  angles  is 
the  least.  Let  AV,  //V,  v'n  be  the  angles  of  this  triangle,  and  let 
X" ,  )x",  v"  denote  the  numbers  A',  /i',  v'  arranged  in  descending  order 
of  magnitude.  The  following  table  gives  the  systems  of  values  of 
V ,  pL" ,  v"  in  order  that  the  general  integral  shall  be  algebraic : 


A" 


I, 

i 

i  " 

11, 

i 

i  i 

III, 

1 

*  i 

IV, 

1 

2 

i  i 

V, 

1 

i  i 

VI, 

2 

i  i 

VII, 

0 

1       1 

3         3 

VIII, 

1 

\  \ 

IX, 

\ 

2         1 

X, 

1 

i  i 

XI, 

2 

3  2 
"5"        a 

XII, 

1 

1         1 

XIII, 

4 

■5" 

i      i 

XIV, 

i 

2  1 
1        3" 

XV, 

3 

1       i 

Double  pyramids. 
Tetrahedron. 

Cube  and  octahedron. 


^  Dodecahedron  and  icosahedron. 


(24)  We  have  neglected  the  intermediate  case  where  the  circle 


274  LINEAR  DIFFERENTIAL  EQUATIONS. 

described  upon  OC  as  diameter  is  tangent  to  the  perpendicular  SS'  ; 
that  is  to  say,  the  case  where  the  two  homographic  transformations, 

{A)  p'  —  a  —  K{p  —  a), 

have  a  common  double  point.     The  double  points  of  the  first  are 

^  =z  a  and  ^  ^  oo ;  those  of  the  second  are  ^  =  o,  S  ^  -7-.     If  we 

refer  to  the  values  of  a  and  d,  which  may  become  zero,  but  not  in- 
finite, we  can  see  that  there  are  three  different  ways  in  which  the 
two  homographic  transformations  may  have  a  common  double  point, 
(i)  a  =  o;  this  will  happen  when  one  of  the  numbers  a -\-  1  —  y, 
^ -\-  I  —  y  is  zero  or  a  negative  integer. 

(2)  d  =  0;  this  will  happen  ii  y  —  a  or  y  —  ^  is  zero  or  a  nega- 
tive integer. 

(3)  al?  :=  I  ;  this  condition  gives 

sin  {y  —  a)7r  sin  [y  —  ft)7t  ^  sin  yn  sin  {y  —  a  —  I3)7t, 

or 

cos  {a  —  /5)7r  =  cos  {a  -\-  0)7t, 

from  which 

{a-  I3)±{a  +  ^)=2m; 

one  of  the  numbers  a,  13  must  then  be  an  integer.  In  order  to  see 
what  takes  place  in  each  of  these  cases,  suppose  the  common  double 
point  at  infinity  ;  the  homographic  transformations  will  be  defined  by 

z'  —  z,  —  K{z  ~  z^), 
z'  -z,  =  K'  {z  -  z,), 

the  points  z^  and  z^  being  the  other  two  double  points.  These  give 
rise  to  a  simple  geometrical  construction.  Having  given  in  the 
plane  a  point  M  representing  z,  we  shall  find  the  point  M'  repre- 
senting z'  by  turning  the  radius  z^M  through  an  angle  gj  round  z^, 
or  the  radius  z^M  through  an  angle  ay'  round  z,  .  Starting  from  any 
point  of  the  plane,  it  is  clear  that  we  can  apply  the  constructions 


GOURSAT:  HYPERGEOMETRIC  SERIES.  275 

■successively  in  such  an  order  that  we  shall  never  arrive  at  a  point 
already  found  ;  for  example,  we  can  arrange  so  that  in  following  the 
process  the  radius  z^M  never  decreases.  The  common  double  point 
is  a  case  of  exception,  as  it  will  always  coincide  with  itself.  There 
will  then  be  a  particular  integral  whose  logarithmic  derivative  has 
only  a  single  value  in  each  point  of  the  plane  ;  this  derivative  is 
therefore  a  rational  fraction.  Under  the  adopted  hypothesis,  i.e., 
where  the  numbers  a,  [3,  y  are  real  and  rational,  the  corresponding 
particular  integral  will  be  an  algebraic  function.  We  see,  further, 
that  there  is  no  other  such  integral.  We  must,  however,  remark 
that  this  will  cease  to  be  true  if  the  other  two  double  points  are  the 
■same  ;  in  this  case  all  the  integrals  will  be  algebraic.  This  case 
arises  when  we  have  simultaneously  «  ^  o,  <^  =  o. 
For  example,  consider  the  differential  equation 

s  d'^y       ,  s  dy 

where 

a  =  /3  =  h    r  =  l- 

The  equation  admits  a  particular  algebraic  integral  j/  =  — ^- ,  but  the 


general  integral, 


C         C'  sin-'  Vx 


V  X  \  X 

is  transcendental.     On  the  contrary,  take  the  equation 


Ahere 


X  ^>   ,    (^        -n  dy       20 


We  have  simultaneously  a  ^=  o,  b  =  o ;  the  equation  admits  the  two 
particular  integrals, 

ex\ 


xl[x  -— j,     {i-x)l 


and  so  the  general  integral  is  algebraic. 


276  LINEAR   DIFFERENTIAL   EQUATIONS. 

(25)  In  what  precedes  we  have  supposed  the  two  integrals  0,  and  0^ 
to  be  distinct ;  i.e.,  neither  of  the  numbers  a,  /?  is  zero  or  a  negative 
integer.  If  this  circumstance  does,  however,  arise,  we  can  replace 
03  by  another  integral,  for  example  0^, ,  and  operating  as  above,  we 
shall  be  led  to  consider  two  homographic  transformations.  These 
two  transformations  will  have  a  common  double  point,  since  equa-  \ 
tion  (i)  admits  as  an  integral  an  entire  function  whose  logarithmic 
derivative  is  a  rational  fraction.  If  the  other  two  double  points  are 
different  (which  will  generally  be  the  case),  there  will  not  be  any 
other  algebraic  integral.  But  if  these  two  double  points  coincide,, 
the  general  integral  will  be  algebraic. 

In  order  that  this  may  be  so,  one  of  the  elements  a,  /?  must  be 
zero  or  a  negative  integer,  and  the  other  a  positive  integer. 

Thus  the  differential  equation 

,  d'^y       I  dy 

where 

«  =  —  2,     /?  =  I,     Y  =  \^ 

admits  two  particular  algebraic  integrals,  and  therefore  the  general 
integral  will  be  algebraic,  viz. : 


Part  Second. 

(I)  The  memoirs  of  Gauss  and  Kummer  on  the  hypergeometric 
series  contain  a  great  many  formulae  not  found  among  those  given 
above,  and  which  only  exist  when  the  constants  a,  yS,  y  satisfy  cer- 
tain conditions.     The  general  type  of  these  formulae  is 

;t-/(i  _  xy<JF{a,  /?,  y,  X)  =  t^' {x  -  ty F{a\  yS',  y\  t\ 
where  t  is  an  algebraic  function  of  x.     The  function 
xf{\-  xytP'{\  —  tyF{a',  ft',  /,  t) 


h 


GOURSAT :   HYPERGEOMETRIC  SERIES.  277 

is  then  an  integral  of  the  differential  equation 

(I)  x{l  -  x)-£^^\y  -{a^  fi  ^  l)x\-^^-  afty  :=.0. 

We  are  tlius  led  to  seek  the  cases  in  which  equation  (i)  admits 
integrals  of  the  above  form.  Such  is,  very  nearly  at  least,  the  path 
followed  by  Kummer.  Kummer,  however,  indicates  no  means  of 
finding  all  the  cases  where  such  integrals  exist.  This  question  we 
propose  to  treat  in  what  follows. 

(2)  We  will  adopt  a  slightly  different  point  of  view  from  that  of 
Kummer.  Let  us  denote,  as  Riemann  does,  by  Pipe)  a  non-uniform 
function  of  x  possessing  the  following  properties  : 

(i)  It  admits  in  the  entire  extent  of  the  plane,  or  of  the  sphere, 
only  three  critical  points,  viz.,  jir  =  o,  ;r  ==  i,  ;ir  =  00  ;  it  is  holomor- 
phic  in  any  region  of  the  plane  having  a  simple  contour  which  does 
not  contain  either  of  the  points  x  =^  o,  x  ^^  i. 

(2)  Between  any  three  branches,  P\  F' ,  P'" ,  of  the  function 
there  exists  a  linear  homogeneous  relation 

CF  4-  C"F'  +  C"F"'  =  o 

with  constant  coefficients. 

(3)  Each  branch  of  the  function  is  finite  for  x  ^=  o,  x  ■=  i,  and 
also  for  ;ir  =  00  when  we  multiply  it  by  a  proper  power  of  ;t-  or  i  —  x. 

Riemann  has  shown  that  certain  branches  of  the  function  P  can 
be  expressed  by  products  such  as;tr"^(i  —  x)~'^F{a,  /3,  y,  x).  In  the 
light  of  the  more  recent  analysis,  this  can  be  demonstrated  by  a 
simpler  method  than  that  adopted  by  Riemann.  It  results,  in  fact, 
from  a  theorem  given  by  Tannery,*  viz. : 

T/ie  different  branches  of  the  function  P  are  integrals  of  a  linear 
differential  equation  of  the  second  order  having  uniform  coefficients, 
and  having  no  other  critical  points  than  the  points  0,1,  00  /  further, 
all  these  integrals  are  regular  in  the  region  of  a  critical  point. 

This  differential  equation,  as  Fuchs  has  shown,  is  of  the  form 

d'^P  dP 

<2)       X\l   -  ;.y  -^+  [/  -  (/+  m)x-\x{l   -X)^ 

+  {Ax'  +  Bx+  C)P  =  o. 

*  Annales  de  I'Ecole  normale  supe'neure ,  2"  s6rie,  t.  iv.  p.  130. 


2/8  LINEAR  DIFFERENTIAL   EQUATIONS. 

We  pass  from  (i)  to  (2)  by  writing 

y  =L  x*'{\  —  xyP ; 
A,  B,  C,  /,  in  are  given  by  the  formulae 

{C=p{p-l-\-y\   A^B^C^qicj^a^fi-  y). 
Conversely,  we  pass  from  (2)  to  (i)  by  making 
P  —  x-\\  —  xy^; 

a,  f3,  y,  p,  q  will  now  be  determined  by  aid  of  equations  (3)  in  terms; 
of  A,  B,  C,  I,  m.  Riemann's  theorem  can  be  established  in  the  same 
way. 

[In  order  to  completely  define  the  function  P  it  is  necessary  to 
add  the  following  condition,  viz. :  if  P',  P"  are  two  Hnearly  distinct 
branches,  the  determinant 


Z>  = 


P,       P" 

dP      dP" 
dx  '      dx 


must  be  difTerent  from  zero  for  every  point  of  the  plane  other  than 
the  points  ;t:  =  o,  ;f  =  i.  If,  in  fact,  this  determinant  vanished  for 
X  ^=.  a,  the  point  a  would  be  an  apparent  singular  point  for  the  differ- 
ential equation.  Consider,  for  example,  the  integrals  of  equation 
(i) ;  it  is  clear  that  they  satisfy  the  conditions  which  serve  to  define 
the  function  P.  At  first  sight,  the  same  is  true  of  the  products  ob- 
tained by  multiplying  each  of  these  integrals  by  the  factor  {x  —  2) ; 
nevertheless,  these  new  functions  satisfy  the  differential  equation 


x{i  —  x){x  —  2)- 


(Pz 
Tx^ 


2V 


dz 


+    ■l[r-(«  +  /5+   l)x'\{x-2)-   2X{1-X)\{X         ^,^^^ 

-^  \2x{i-x)+[{a-\-  ^-^  l)x  -y-]{x-2)-  aft{x  -  2Y 


=  0, 


which  is  not  comprised  in  the  form  (2) ;  but  the  determinant  D  is 
evidently  zero  for  the  point  x  =  2.] 


GOURSAT:  HYPERGEOMETRIC  SERIES.  279 

Remark  that  having  given  a  system  of  values  for  A,  B,  C,  I,  m, 
there  result  four  systems  of  values  for  a,  /?,  y,p,  g!\  equation  (2), 
and  consequently  equation  (i),  admits  then  four  integrals  of  the  form 
x'\i  —  x)''''F{a,  /?,  y,  x) — a  well-known  result. 

Let  /  be  a  new  variable  given  by  the  equation  x  =  <f){t).  If, 
when  in  equation  (2)  we  change  the  variable  by  the  relation  x  =  <p{t), 
the  function  P  satisfies  the  same  relations  as  relatively  to  x,  we 
ought  to  find  a  new  differential  equation  (4)  analogous  to  (2),  viz. : 

d"P  dP 

(4)     f{x-ty^^^  [/■■  -  (/'  +  m')t\t  {i-t)-^ 

^{A'f  +  B't  +  C')P=o. 

It  is  easy  to  see  that  the  problem  treated  by  Kummer  is  herein 
contained  ;  if,  in  fact,  equation  (i)  admits  the  integral 

xP{i  —  x)U^'{l  —  tYF{a',  13',  y',  t), 

equation  (2)  will  admit  the  integral 

t^'{i~-tYF{a',l3',y',t). 

If  then  in  equation  (2)  we  change  the  variable  by  the  relation 
X  =  (p{i),  we  must  obtain  an  equation  of  the  form  (4). 

The  problem  to  be  studied  may  now  be  stated  as  follows : 

Required  to  determine  for  what  values  of  the  constants  A,  B,  C,  /, ;«, 
there  exist  transformations  such  as  x  :=  cj){t),  for  which  equation  (2) 
does  not  change  its  form,  A,  B,  C,  I,  m  being  simply  replaced  by  the 
nezv  constants  A' ,  B' ,  C,  I',  ni' . 

When  we  have  found  the  conditions  which  the  constants 
A,  B,  C,  I,  m  must  satisfy,  equations  (3)  will  give  us  the  conditions 
to  be  satisfied  by  the  elements  a,  ft,  y  themselves. 

(3)  In  equation  (2),  making  the  change  of  variable  x  =  0(^),  and 

dx  d"^  X 

writing  x'  =  -77,  x"  =  —j:r,  we  have 
^  dt  dt 

dP_   I  dP 

dx       x'  dt  ' 

d'P       I   r  .d"-P        ..dP 


dx' 


_  ^  r '  ^  _  -^1 

~  x'^L^  df     ""  dt  y 


280  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  the  new  equation  is 


x\\-  xy  d'P       r/  -  (/ 4-  m)x         _  x"x'{i  -x) 

x"  df  +  L  x'  -^^^  -  ^)  -  ^'^ 


dP 

dt 

-^{Ax'  +  Bx^C)P=o, 


where  x,  x' ,  x"  are  to  be  replaced  by  their  values  as  functions  of  /. 

In  order  that  the  new  equation  shall  have  the  desired  form,  we 
must  have 


(5^  x\i-x)\     _    f{i-tr 


x" {Ax'  -\-Bx  -^C)~  A'f-\-  B't  +  a ' 

l-{l^m)x    ,_£:'_  I'  -{l'-^m')t 
^^  x{i  -  X)  x'   ~        t{l  -t)        ' 

This  last  equation  can  be  integrated  once,  since  it  can  be  written  in 
the  form 

J,  log  [^(.  -  -)■"]  -  ^,  log  y  =  j^  log  [/'(I  _  0"']. 

We  can  then  replace  equations  (5)  and  (6)  by  a  system  of  two  differ- 
ential equations  of  the  first  order,  viz. : 


VAx'  +  Bx  +  C    ,  VA'f-\-B'i  +  C  , 

(5)  r T ^^  = TT :^ ^^f 

^^^  x{i  —  x)  t{i  —  t)  ' 

,  ^  dx  Kdt 

(7) 


x^{\  —  xY    t^'{\  —ty 

The  constants  y^',  B' ,  C ,  I' ,  m',  K  are  arbitrary,  and  we  have  now     I 
to  see  in  what  cases  we  can  determine  them  so  that  equations  (5) 
and  (7)  shall  have  a  common  integral. 

(4)  The  following  five  transformations  are  readily  seen  to  exist 
in  all  cases,  whatever  be  the  values  oi  A,  B,  C,  /,  m  ;  viz.  : 

_  I  _      I  _      /  _  ^  ~~  ^ 

x=i-t,     X  -~,     X  -  YZ^^ ,     X  -  j-—^  ,     X  -  —J-  . 

This  result  is  also  evident  if  we  refer  to  the  definition  of  the  func- 
tion P.      If  a  multiple-valued  function  satisfies  the  required  condi- 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


281 


lions  when  we  take  x  as  the  variable,  it  is  clear  that  it  will  also 
satisfy  these  conditions  when  we  take  as  variable  one  of  the  quanti- 

II  XX  —  I 

ties  I  —  X ,  —  ,  ,   ,  ;  for,  the  values  of  any  one  of 

X      \—x     x—\         X 

these  quantities  for  ;tr  =  o,  I,  co  will  also  be  o,  i,  00  taken  in  a  cer- 
tain order.  From  these  considerations  arises  a  very  simple  method 
for  finding  Kummer's  twenty-four  integrals,  but  we  will  not  take  up 
that  point  here.* 

We  can  show  now  that  there  can  exist  no  transformation  of  the 
first  order  between  the  two  variables  other  than  the  above-mentioned 
ones.     Consider,  in  fact,  the  transformation 


X  = 


ai  -\-d 
a  -\-d' 


If  the  values  of  t  corresponding  to  the  values  x  =  o,  x=i,x=co 
are  also  o,  i,  00,  taken  in  a  certain  order,  the  transformation  will  ob- 
viously be  one  of  the  preceding  ones.  Suppose  on  the  contrary  that 
for  X  =  o,  for  example,  t  takes  a  finite  value  /,  different  from  o  and 
from  I.     Let  x  (Fig.  14)  describe  a  small  loop  round  the  origin  ;  the 


Fig.  14. 


point  /  will  describe  a  small  loop  surrounding  f^ .  After  describing 
such  a  path,  any  integral  whatever  of  (4)  will  return  to  its  initial 
value :  the  point  x  =  o  itself  can  then  not  be  a  critical  point  for 
equation  (2),  which  is  contrary  to  hypothesis. 

We  see  further  that,  if  for  proper  values  oi  A,  B,  C,  I,  in,  we  can 
make  the  transformation  x  =  cp  (/),  we  can  also,  for  the  same  values 
of  the  constants,  make  the  five  other  transformations 

..-=0(1-/),    ^=0(7),    -^=0(7:^,),     ^  =  ^(737)'    ''=^(-^)' 


This  method  has  been  given  in  Chapter  VI. — AUTHOR. 


282  LINEAR  DIFFERENTIAL   EQUATIONS. 

From  the  form  of  equations  (5)  and  (7)  we  can  deduce  still  further 
consequences.  If  for  the  values  of  A,  B,  C,  I,  in  we  can  make  two 
different  transformations  ;f  =  0(/),  x^=^ip{t),  then  for  properly 
chosen  values  of  the  constants  we  can  make  the  transformation 
(p{x)  =^  ip{t).  Thus,  from  a  transformation  x^(p{t)  we  would 
be  able  to  deduce  all  those  which  would  be  obtained  either  in  re- 

II  XX  —   I 

placing  ^  by  I  —  ;ir,  -  ,    -— — - ,   — -—  ,    — — -  ,  or  in   replacing  /  by 

II            t        t  —  \         .  ,  .        , 

I  —  /,  - , , ,  — - — ,  or  m  makmg  the  two  transforma- 

if  X    — ^    ^  I,    •  ^  {, 

tions  simultaneously,  making  thus  in  all  thirty-six  transformations, 
not  all  of  them,  however,  being  different.  We  will  determine,  at  the 
same  time,  all  of  these  transformations,  together  with  the  inverse 
transformations. 

(5)  If  equations  (5)  and  (7)  admit  a  common  integral,  this  integral 
will  also  satisfy  the  equation  obtained  by  dividing  the  equations  (5) 
and  (7)  member  by  member,  viz., 


VAx'-\-Bx-^C.x^-'{i-xy"-^=KVA'f-{-B'^-{-C\i''-ii-ty"'- 
Taking  the  logarithmic  derivative,  we  have 

,  r         2Ax-{-B  4_  ^~  ^    I    ^'^  ~  ^ 

\_2{Ax'  -^  Bx  -\-  C)  ~^        ^  ~^   X-  I. 


dt 


[_2{A'f-\^B't  +C')'^      t      +  /  -  I  _ 


Replacing,  from  equation  (5),  dx  and  dt  by  the  quantities  to  which 
they  are  proportional,  and  squaring,  we  have,  finally, 

(8)        \{2Ax  +  B)x{x  -  I)  +  2{Ax'  ^Bx  ^  C) 

[(/-i)(^--i)+(^^-i>]}' 

{Ax""  -f  Bx  +  cy 

\{2A't-^  Byit-  i)  +  2(^r  +  .57  +  C) 

^ w-  i)(/-i)+K-i)/]r 

{A'f-^B't-\-Cy 


GO  URSA  T:  HYPERGEOMETRIC  SERIES. 


283: 


If  this  relation  is  not  an  identity,  we  see  that  x  and  t  are  connected 
by  an  equation  of  at  most  the  sixth  degree  in  each  of  the  variables^ 

Equation  (8)  will  be  an  identity  if  the  two  members  reduce  to 
zero  or  to  equal  constants. 

I.  In  order  that  each  member  of  (8)  shall  be  identically  zero  it  is. 
necessary  and  sufficient  that 


S/Ax^  -^  Bx  -\-  C  .  x^-  '(i  -  xy 


and 


VA'f  +  B't  +  C .  /''-'(i  -  t)"''-' 


reduce  to  constants;  Ax'' -\-  Bx  -f-  C  =  o  should  then  admit  of  nO' 
other  roots  than  o  and  i,  and  the  same  should  be  true  of 

A'f+B't  +  C'  =  o. 

All  of  the  possible  combinations  are  given  in  the  following  table: 


1st  case. 

2d  case. 

3d  case. 

4th  case. 

5th  case. 

6th  case. 

A-\-B=  0, 

A  =  0, 

A  =  0, 

A  =  C, 

A  =  o, 

^  =  0, 

C=  0, 

C  =  0, 

B^C=o, 

B-2A=o, 

B  =  0, 

C=  0, 

/=  m  =  h 

1=^,  m=i. 

1=1,  m—\, 

/:=!,   W=0, 

/=m=i, 

/=zO,  m=\, 

A  =  ±  i, 

A  =  ±i 

^=±h 

M=  ±   h 

v=  ±   I, 

A  =  ±  I, 

M=  ±  h 

r=±h 

y  =  ±  h 

^  =  ±  y, 

X=±M, 

fX=    ±    V. 

In  the  last  two  horizontal  lines  we  have  given  the  corresponding 
conditions  to  be  satisfied  by  the  elements  a,  /?,  y,  where 


A  =  I 


}x  =.  y  —  a  —  §,       V  =^  ft  —  a. 


The  conditions  for  A' ,  B',  C,  I', ;;«' will  be  identical  in  form  with 
the  above.  In  each  of  these  cases  equations  (5)  and  (7)  reduce  to 
the  same,  and  so  an  infinite  number  of  changes  of  the  variable  can 
be  made  which  will  leave  (2)  unaltered  in  form.  For  example,  take 
A  -{-  B  =  o,  C  =0,  /  =  m  =  ^;  it  is  suiificient  now  to  take  for  x: 


'284  LINEAR  DIFFERENTIAL   EQUATIONS. 

an  integral  of  one  of  the  following  differential  equations,  in  which  K 
is  arbitrary: 

dx  Kdt 


dx  Kdt 


Vx{i  —  x)  tS/\  —  t  ' 

dx  Kdt 


Vx{i  —  x)         I-/'  Vx{i  —  x)         t{i  -  t)' 


It  is  easy  to  assure  one's  self  that,  in  all  these  cases,  the  general 
integral  of  (4),  and  consequently  of  (2),  is  expressed  by  means  of 
•elementary  functions.  Remark  first  that  the  second  and  third  cases 
can  be   conducted   to  the    first    by   changing   respectively  x   into 

X  I 

■ or  into  —  ;  so  also  the  fifth  and  sixth  cases  can  be  led  to  the 

JV  —  I  X 

fourth  by  changing  x  into or  into  i  —  x.     It  remains  then  to 

•consider  the  two  equations 

dT  dP 

X  -^-7-  -\-  x-. \-  AP  =  o. 

dx  dx 

The  general  integral  of  the  first  of  these  is 

P  —  - 

sin-'—  =  VA  sin  -'  Vx  -{-  C,, 

and  of  the  second  is  "^ 

P  =  C.x'-  +  Qx- ", 

^vhere  r  =  VA.     In  particular,  if  ^4  =  o,  the  last  integral  is 

P=C,-\-CJogx. 


GOURSAT:   HYPERGEOMETRIC  SERIES.  28^; 

2.  It  may  happen  that  the  two  members  of  (8)  reduce  to  con- 
stants ;  for  this  it  is  necessary  that  Ax^  -\-  Bx  -\-  C  admits  x  ox  x  —  i 
as  double  factors  or  that  it  reduces  to  a  constant.  The  following 
are  all  the  possible  combinations : 


1st  case. 

2d  case. 

3d  case. 

^  =  o, 

A  =  C, 

A  =  o, 

(7=0, 

B  -  2A    =  o 

B  =  o, 

/  =  o, 

m  =  o, 

I  -\-  VI  =  2. 

We  find  for  a,  /?,  y  the  same  conditions  as  in  the  last  three 
examined  cases. 

Suppose,  for  example,  A  =  C,  B  —  2A  =^o,  m  ^  o\v^q  can  make 
any  one  of  the  transformations 

x  =  Kj'',    x  =  Kii-tr,     x  =  K,[^J, 

where  K^  and  n  are  arbitrary.     It  is  easy  to  see  now  that  the  general, 
integral  of  (2)  is  of  the  form 

P  =  Qx'-  +  C,x-\ 

Summing  up:  When  (8)  reduces  to  an  identity  we  can  effect  an 
infinite  number  of  changes  of  the  variable  in  (2)  so  as  to  leave  the 
form  of  this  equation  unaltered,  and  can  so  deduce  an  infinity  of 
relations  included  in  the  type  of  Kummer's  formulae  ;  but  it  is  to  be 
noticed  that  all  of  these  relations  exist  between  functions  which  are 
expressed  by  means  of  the  elementary  functions,  exponential,  circu- 
lar, or  logarithmic. 

(6)  Discarding  these  particular  cases,  suppose  that  (8)  does  not  re- 
duce to  an  identity.  If  now  there  exists  an  integral  common  to  (5) 
and  (7),  ;r  will  be  an  algebraic  function  of  /  defined  by  an  equation  of, 
at  most,  the  sixth  degree  in  x  and  in  /.  It  is  therefore  only  among 
algebraic  functions  that  we  must  seek  the  functions  which  will  per- 
mit us  to  transform  (2)  in  the  desired  manner.  We  shall  determine 
first  the  rational  transformations,  and  after  will  show  that  all  the 
other  transformations  can  be  conducted  to  these. 

P 

Let  ;ir  =  -^  be  a  rational  transformation  ;  R  and  5  are  two  poly- 


2S6  LINEAR  DIFFERENTIAL   EQUATIONS. 

nomials  of  degrees  at  most  =  6,  and  at  least  one  of  them  is  of  a 
degree  higher  than  unity.  Among  the  values  of  /  which  correspond 
to  the  values  O,  i,  oo  of  ;r  there  will  be  at  least  one  which  is  differ- 
ent from  o,  I,  00.  Suppose,  for  example,  that  for  ;r  =  o  we  have 
/  =  O,  and  for  x  ^^  co,  /  =  i  ;  then  R  will  be  equal  to  Kv\  and  5  to 
K\\  —  /)'.  The  values  of  /which  correspond  to  ;f  =  i  will  then  be 
the  roots  of  the  equation  KV  —  K'{\  —  t)'  =  o.  One,  at  least,  of 
the  numbers  r,  s  being  greater  than  unity,  the  left-hand  member  of 
this  equation  cannot  reduce  to  a  constant;  further,  the  equation  does 
not  admit  either  o  or  i  as  a  root.  There  are  therefore  finite  values 
of  /  which  are  neither  o  nor  i  and  which  correspond  to  ;ir  =  i.  Sup- 
pose then  that  for  x  =:  o,  for  example,  /  takes  a  value  /,  which  is 
neither  zero  nor  unity.  The  point  x  =^o  will  be  a  critical  point  for 
that  value  of  /  which  becomes  /, .  In  fact,  let  the  variable  x  describe 
a  small  loop  round  the  origin  (Fig.  14):  if  /were  a  holomorphic 
function  of  x  in  the  region  of  the  point  ;ir  =  o,  it  would  return  to  its 
original  value  after  having  described  a  small  loop  round  /, ,  and  we 
could  then  conclude,  as  above,  that  the  origin  would  not  be  a  criti- 
cal point  for  equation  (2),  which  is  not  the  case.  It  is  therefore 
necessary  that  several  values  of  /  become  equal  at  /,  when  x  =  o. 
Suppose  71  to  be  the  number  of  these  values;  then 

^  =  (^  -  ^>)Y(0. 

y(/)  being  a  rational  function  of  /  which  is  neither  zero  nor  infinite 
for  /  =  /, .     In  the  same  way  we  shall  have 

^  =  (.-,)"-/,«, 

/^(t)  possessing,  relatively  to  the  point  /, ,  the  same  properties  as 
_/(/).     From  equation  (5)  we  have,  further. 


dx  ._  x{i-x)  VA'f  +  B't-\-  C 
dt    ~   /(I  -  /)  ^Ax-"  -^  Bx  -\-  C' 


I  —  X 


The  quotient  —, — r   is  different  from  zero  for  t  =^  t^\  if  neither 

/(I        t) 

Ax""-]-  Bx  -\-  C  nor  A'f  -\-  B't  -j-  C  were  zero  for  ^r  =  o  and  t  =  t^y 


G  OUR  SAT:  HYPERGEOMETRIC  SERIES.  28/ 

Ave  should  have 

ip{t)  being  a  uniform  function  of  t  in  the  region  of  the  point  /,  and 
different  from  zero  for  t  =  t^,  which  is  impossible.  If  A'f-\-B't-\-C' 
were  zero  for  /  =  /,  and  C  not  zero,  we  should  have 

dx  dx 

^=(t-  ty + Hit),  or  5^  =  ('  -  ',)• + ■  m, 
I 

ip{t)  having  the  same  meaning  as  above.  This  is  also  impossible.  We 
therefore  conclude  that :  If  for  x  =  o,  t  takes  a  finite  value  t^  tvJiich 
is  neither  zero  nor  unity,  it  is  necessary  that  the  constant  C  be  zero. 

(7)  It  remains  to  find  the  values  which  the  integer  n  may  take. 
Several  cases  are  to  be  examined,  as  follows  : 

First  hypothesis  : 

A't:+B't,-\-C'<o,     B<o; 
then 

dx  ,-     _         ,  ,  ^     .  , 


dt    ~ 

-      T 

■^  'hV)  —  V  — 

'1;  f 

iVJ ' 

also, 

' 

dx 
~dt    ^ 

--{t 

-  trvii)- 

We  must  have. 

therefore. 

n 

2 

-  n 

—  I,     whence 

71  = 

2. 

Second  hypothesis : 

A't, 

=  + 

B't,-\-C'^o, 

B 

=  0; 

consequently 

dx  ,  .  , 

—  =  ^•(/),     whence  n  =  i. 

This  hypothesis  is  therefore  to  be  rejected. 


288  ■  LINEAR  DIFFERENTIAL   EQUATIONS. 

Third  hypothesis: 

A't:^  B't,-\-  C  =  o,     2A't,-\- B'<o,     ^<o; 

it  follows  that 

dx  ^±i  n  -\-  \ 

—  —{t  —  t^  ^  ip{t),     whence—;^  =  n—  \,  n  =  2,. 

Fourth  hypothesis  : 

A't,'-\-B't,+  C  =  0,     2A't,^B'<o,     B  =  0] 
which  gives 

—  =  (/  —  /j)i  ^■(/),     whence  «  =  |. 

The  transformation  being  supposed  rational,  this  hypothesis  is  to  be 
rejected. 

Fifth  hypothesis  : 

A't;'^  B't,-{-  C  =0,     2A't,-\-  B'  =  o,    B<o; 
therefore 

from  which  we  derive 

n 

— \-i  ^=n  —  I,     n  =  A. 

2 

Sixth  hypothesis  : 

A't;-{-B't,-\-C'=o,     2A't,+  B'=o,    B  =  o; 

then 

dx       ,  ...  , 

-7-  =  (/  —  t^)il\t),     whence  11  =  2. 

The  only  admissible  values  for  the  integer  n  are  consequently,  2,  3,  4. 

We  can  find  the  corresponding  values  of  /  by  remarking  that  the 

I    dx  ,       -    .       ^ 

ratio  -y  -J-  must  be  finite  lor  x  ^^  o,  t  =  t^\  this  requires  that  we 

have  /  = . 


GO  URSA  T:   HYPERGEOMETRIC  SERIES.  289 

(8)  We  can  prove  in  like  manner  that  if  /  takes  a  value  different 
from  o,  I,  CO  for  x  ^  i  or  for  x  =  00,  we  must  have  A  -\-  B  -{-  C  ^  O 
in  the  first  case,  and  A  =  o  in  the  second.     To  verify  this  it  is  only 

necessary  to  change  x  into   i  —  x  for  the  first  case,  and  into  —  for 

the  second.  Discarding  the  special  case  where  we  have  simul- 
taneously ^  =  ^  =  C  =  o,  we  find  that,  if  we  give  x  the  values 
o,  I,  00,  there  is  at  least  one  of  these  values  for  which  none  of  the 
values  of  /  is  different  from  o,  i,  co  ,  Suppose  the  case  ;ir  =  00  ,  and 
suppose  further  that  for  x  =  o,  ^  takes  a  finite  value  /,  which  is 
neither  o  nor  i  ;  we  shall  always  have  C  ==  O.  There  are  two  cases 
to  distinguish  according  as  among  the  values  of  t  corresponding  to 
X  ^  I  there  are  or  are  not  values  which  are  neither  o,  i,  00 . 
The  transformation  will  have  one  of  the  following  forms : 

(^) 
(^) 
(0 
(d) 

ie) 
(/) 
{g) 
{h) 

R  denoting  an  entire  function  which  has  no  double  factor,  and  which 
is  not  zero  for  either  /=  o  or  /  =  i  ;  n  is  one  of  the  numbers  2,  3,  4, 
and  r  and  s  are  positive  integers.  We  can  reduce  the  number  of 
these  transformations.     Thus,  we   can   suppress  the  forms  {c),  {/), 


X 

= 

^'7' (I  - 

-0^ 

X 

= 

R-r, 

X 

= 

R"{i  - 

OS 

X 

= 

R", 
R"r 

X 

(I  -  ty 

) 

= 

R"{i- 

ty 

X 

r 

J 

R" 

X 

R'^ 

X 

— 

{i-ty 

> 

R" 

I 


290  LINEAR  DIFFERENTIAL   EQUATIONS, 

{Ji),  which  by  changing  /  into  i  —  t  are  conducted  to  the  forms  {h), 

(^)'  (a")-     Take  now  the  transformation  {g)  and  let  r'  be  the  degree 

I    R" 
of  the  numerator.     If  r'  ^  r,  then,  by  changing  t  into  -,  —7  changes 

into  -7^  or  into  R^f  '"',  and  we  are  thus  conducted  to  the  form  {U) 

t  —  I    R" 
or  the  form  {d).   If  r'  >  r,  then  on  changing  t  into ,  -^  changes 

R" 
into   — ; — 7-^ — iwj  and  so  we  are  conducted  to  the  form  (t).     Take 
r  -''(i  — //  ^  ' 

now  the  form  (r)  and  let  s'  be  the  degree  of  the  numerator.  If 
s    "^  s,  changmg  /  into changes  -, 7-  mto 

R"r 

(I  -  ^y-'        '    ^       ^    ' 

and  we  are  conducted  to  the  form  (a)  or  to  the  form  (d).     If  s'  >  s, 

I  R"r      .  i?.«  ,        , 

changing  /  into  -  changes  .  _^y  mto  ^,/_,.  _  ^y,  and  so  the  trans- 
formation is  conducted  to  the  form  (/).  We  are  therefore  confined 
to  considering  the  four  following  forms : 

(a)  X  =  R"r{i  -/)% 

{b)  X  =  R"t\ 

{d)  X  =  R", 

_       R" 

(9)  We  now  proceed  to  calculate  the  unknown  coefificients  which 
enter  into  these  transformations.  Suppose  first  that  for  ;ir  =  I  there 
is  no  value  of  /  different  from  o,  i,  co.  If  the  transformation  has 
the  form  {a),  the  values  of  /,  for  ;ir  =  i,  are  given  by  the  equation 

R"r{i  —  /y  —  I  =0. 


GOUA'SAT:   HYPERGEOMETRIC  SERIES.  29 1 

It  is  clear  that  this  equation  does  not  admit  either  /  =  o  or  /  =  i  as 
roots,  and  that  the  first  member  does  not  reduce  to  a  constant. 
Take  in  the  same  way  the  form  {b) ;  the  equation  RPV  —1=0  must 
only  admit  of  the  root  /  =  i  ;  consequently 

72«r=  I  +//(i  -ty. 

Now  the  first  member  of  this  admits  multiple  factors,  while  the 
second  does  not ;  the  equality  is  therefore  impossible.  If  we  take 
the  form  {d),  the  equation  ^"—1=0  ought  to  admit  no  other 
root  than  o  and  i  ;  this  requires  that  n  be  equal  to  2,  and  that  R  be 
of  the  first  degree.  It  is  easy  to  see  that  we  must  take  R=^2t—  i  ; 
from  this  results  the  transformation 

X  =  {2t  —  i)\ 

For  the  form  (/)  we  must  have  that 

7?«  —  r(i  —  ty 

reduces  to  a  constant  H.  The  equation  R"  —  H  =  o  cannot  admit 
/  =  O  or  /  =  I  as  roots.  We  are- thus  led  back  to  the  preceding  case, 
which  gives  now  the  transformation 


4t{t  -  I)' 


In  order  that  these  transformations  may  be  effected  we  must  have 
C=o  and  /=  i-  Following  is  the  table  of  transformations,  deduced 
by  the  above  process,  together  with  the  inverse  transformations.  On 
the  left-hand  side  of  the  table  are  given  the  conditions  to  be  satis- 
fied by  the  constants  A,  B,  C,  I,  m  and  the  elements  a,  /3,  y  them- 
selves.    The  quantities  A,  ^,  v  have  the  same  meaning  as  before. 


C  =  o, 
/  —  J- 

''    —    2' 


i....  =  i2t-.r,.-^(i^]\x  =  ('+^ 


t    y '  -^  -  Vi  _  /y ' 


'  •*■  =  -7 \ ,  •*■ 


4</-i)'  4(1-0'  4^ 


292 


LINEAR  DIFFERENTIAL  EQUATIONS. 


A+B+C=o, 


in  =  i, 


III...  X  =  4^1  -t),  x  =  ^^^-^,  X  =  ^^  J^ 


A  =  o, 
I  -\-  m  =  I,    ' 


IV...  X  = 


,  X 


Y...X 


4/(1-/)'  4(^-0' 

t 


X  = 


(2/ -I) 


7,,  X 


2   -  / 


,    X 


^■^••••^-^2/-l/'  (2-//' 


(l  -  /)' 

-4/  ' 

i  +  ^r 


^  +  ^  =  o, 
/  =  in, 


I  X  =  ±  /^. 


VII... 


2  2       '  2  V/ 


I  +  i^i  —  /  V/  —  I  +  1// 

X  = ,   X  — 


2VI  —  t 


2Vt  —    I 


X  = 


Vt  -\-\/t  —  I 


2  V't 


VIII... 


4  Vt{t  —  i) 


X 


4V1  -  t 

4Vt 


^  + 

(Tr 

=  0, 

/  +  : 

;«  : 

=  2,  ^ 

;^= 

± 

I'. 

IX... 


;f  = 


,  x  = 


,  x  = 


2  V} 

2  Vt  —  I 


2  1^1   —  / 


.r 


1  +  4/1  —  / 


,  -^ 


2  V? 


1//  +  |//  —    I 


X. 


JIT  -: ZV^  <     ^v    


(4//  +  1//-  i/  (r+Vi-/)' 

4  V't 


X  = 


"(I  +  vty 


GOURSAT:  HYPERGEOjMETRIC  SERIES. 

V}  —  I 


A  -  C  =  o, 

2/  -j-  m  =  2,  - 


V}+i' 


X  = 


Vi—t  —  i 


293 


i+vy 


XI...  ^  x  = 


I 


Vi  —  t 


Vt—  Vt  —  I 


X  =  ±  V. 


I  -\-Vi  —  t 


Vt-\-Vt  -  I 
Vt  —  I  —  V7 


;ir  = 


Vt  ~  i-\-Vt 


XII. 


\  17  —   1  —  17/  \I  —  V  I  —  // 


.ar  = 


i^+  Vt 
1  —  ft 


The  transformations  VII,  IX,  XI  are  the  inverses  of  the  rational 
transformations.  Transformations  VIII,  X,  XII  are  obtained  by 
combining  transformations  I  and  II  in  the  manner  already  ex- 
plained. They  can  also  be  obtained  by  combining  III  and  IV  or 
V  and  VI. 

The  preceding  table  contains  all  the  transformations  given  by 
Kummer  in  the  case  where  two  of  the  three  elements  a,  /3,  y  are 
•arbitrary. 

(10)  Suppose  now  that,  for  .ar  =  i,  several  values  of  /  are  different 
from  o,  I,  00  ;  we  shall  have  simultaneously  C  ^^  o^  A  -\-  B  ^=-  o.  Ex- 
amine now  the  corresponding  forms  of  transformation.     Let 

X  =  R"t\\  —  t)'; 

the  values  of  t  for  x  =  i  are  given  by  the  equation 

R"t'-{i  -  ty—  I  =  o, 

admitting  neither  o  nor  i  as  roots.     We  must  then  have 

R^fii  -  ty  =  I  +  S"', 

where  5  is  an  integral  function  of  the  same  nature  as  R;  n  and  n' 
are  each  one  of  the  numbers  2,  3,  4,  and  r  and  s  are  positive 
integers.    Remark  further  that  each  of  the  members  of  this  equality 


294  LINEAR  DIFFERENTIAL   EQUATIONS. 

must  be  of  a  degree  less  than  or  at  most  equal  to  the  sixth,  and 
that  we  cannot  suppose  n  ^=  n'  ^  2',  for  we  would  have  at  the  same 
time  /  =  7)1  =  ^,  and  should  thus  be  led  back  to  a  particular  case 
already  examined.  These  remarks  made,  we  shall  now  demonstrate 
the  impossibility  of  the  above  equality.  We  see  at  first  that  5  can- 
not be  of  the  first  degree,  for  then  the  second  member  could  have 
no  multiple  factor,  whereas  the  first  member  has  such  factors.  Let 
us  assume 

5  ==  af"  -\-  bt  -\-  c,     ;/'  =:  2  ; 

?i  will  be  equal  to  3  or  4,  and  the  first  member  will  be  of  a  degree 
higher  than  the  fourth.      Let 

S=.af+bt  +  c,     ;^'=3; 

both  members  will  be  of  the  sixth  degree ;  the  second  member  ad- 
mits a  multiple  factor  of  order  of  multiplicity  at  most  =  2.  The 
same  cannot  be  true  of  the  first  member,  for,  if  R  is  of  the  first 
degree  and  11  =  2,  one  at  least  of  the  integers  r,  s  will  be  greater 
than  unity.     Suppose,  finally, 

S  =  at'-^  bf  ^ct^d,     71  =  2; 

71  will  be  equal  to  3  or  to  4,  and  the  first  member  will  admit  either  a 
quadruple  factor  or  a  triple  and  a  double  factor.  The  second  mem- 
ber cannot  admit  a  quadruple  factor,  neither  can  it  admit  a  double 
and  a  triple  factor;  in  fact,  the  double  factor  would  have  to  be  either 
/  or  I  —  /.  Suppose  it  to  be  /  ;  we  must  then  have  c  =  0,  d=^  i,  but 
in  such  a  case  the  equation 

af-^bt''  +  2  =  o 

could  have  no  triple  root. 

Examine  in  the  same  way  form  [d).     In  this  case  we  must  have 

R"  -  I  =  S"'r{i  —  ty. 

This  equality  is  identical  with  the  preceding  one,  save  that  here  r 
and  s  may  be  zero.  If  R  is  of  the  second  degree  and  ;/  =  2,  we 
should  have  ;r  =  3  or  71'  =  4,  and  then  the  second  member  would 


GOURSAT:   HYPERGEOMETRIC  SERIES.  295 

admit  a  triple  or  a  quadruple  factor,  while  the  first  member  has  no 
such  factor.  The  remainder  of  the  discussion  is  the  same  as  above, 
and  we  can  show  in  the  same  way  that  the  preceding  equality  is 
impossible. 

In  order  that  the  form  {b)  may  hold  we  must  have 

iev—  I  =  5"'(i  —  cy, 

where  s  may  be  zero.  If  .y  is  zero,  we  are  led  to  the  preceding  case. 
Suppose  then  r  and  s  not  zero.  The  functions  R  and  .S  will  be  at 
most  of  the  second  degree;  and  as  one  of  the  numbers  «,  n'  must  be 
greater  than  2,  it  is  necessary  that  at  least  one  of  the  functions  be 
of  the  first  degree.     Let 

R  =  at  +  b; 

R'T  —  I  can  only  have  one  multiple  factor,  and  that  must  be  a 
double  factor.  In  fact,  every  multiple  factor  must  be  a  divisor  of 
the  derivative,  that  is,  of 

R""  -  V  -  \nat  +  r{at  +  ^)]  ; 

the  only  suitable  root  of  the  derivative  is  given  by  the  equation  of 
the  first  degree 

nat  -\-  r{at  -f-  <5)  =  o  ; 
further,  this  root  does  not  annul  the  second  derivative 

n{n  —  \)a^f  +  '2nrat{at  -\-b)-{-  r{r  —  i){at  -f  b)'  =  o. 

From  the  first  we  get 

at        at  -\-  b 
r    ~     —  n    ' 

and  substituting  in  the  second  we  find 

—  nr{r  -j-  ;/), 

which  is  always  different  from  zero,  since  r  and  n  are  positive 
integers.  S"'{i  —  t)'  can  then  only  admit  one  double  factor  ;  now, 
if  n'  T=  2,  n  is  greater  than  2,  and  the  first  member  is  at  least  of  the 


296  LINEAR  DIFFERENTIAL  EQUATIONS. 

fourth  degree.     Then  S  should  be  of  the  second  degree  or  ^  >  i  ;  in 
both  cases  the  second  member  admits  more  than  one  multiple  factor. 
The  only  form  which  can  hold,  therefore,  is  the  form  {i), 

X  = 


r{i-ty' 

The  values  of  t  for  x  =^  i  can  be  neither  o  nor  i,  and  they  must  all 
be  roots  of 

i?"  —  r{i  —  ty  =  o 

of  the  same  degree  of  multiplicity.     We  are  thus  conducted  to  the 
following  problem  : 

Required  to  find  two  entire  functions  R  and  S,  and  two  integers  n 
and  n',  so  that  the  equation 

R"  -  S"'  =  o 

admits  only  the  ttvo  roots  o  and  1 ;  R,  S,  n,  n'  being  subjected  to  the 
restrictions  already  indicated. 

As  shown  above,  we  cannot  suppose  71  ==  n'  =z  2 ;  neither  can  we 
suppose  ;/  =  ;?'=  4,  because  the  equation  R*  —  S*  =  o  always  ad- 
mits more  than  two  distinct  roots.  The  only  admissible  hypotheses 
are,  supposing  n  ^  n', 


n  =  I, 

n  =  2, 

n  =  2, 

n  =  3 

n'=l, 

n'=A, 

n  —  3, 

7l'  =  4 

(i  i)  Let  ;?=«'=  3 ;  the  equation  i?^  —  5'  =  O  is  equivalent  to 
the  three  equations 

R  =  S,         R  =jS,         R  =fS. 

The  first  member  of  one  of  these  must  reduce  to  a  constant,  and  the 
other  two  ought  each  to  admit  a  distinct  root :  this  requires  that  they 
shall  be  of  the  first  degree.     Let 

R  =  t  -\-  u,         S  =  t  -\-v; 
we  must  have 

u  =yV,         I  +  u  =j  ^jv, 


GOURSAT:  HYPERGEOMETRIC  SERIES.  297 

from  which 

v=j\         u=j. 

In  fact,  we  have  the  identity 

and  we  deduce  the  transformation  , 


X  = 


iju-^y{^-ty 


The  following  are  the  transformations  which  conduct  to  this  form 
J  is  of  course  one  of  the  imaginary  cube  roots  of  unity : 


'  A  -^  B  =  0,  C  =  o, 

XIII...  ^  /=;;^  =  f, 


A  =   ±  i,   /f  =   ±  i. 

A  =  o,  C  =  o, 
XIV...  ^  /=w=f, 


r       ^  =  T7 


(^  +jy 


5jU-  iKi  -0' 


XV...  ^ 


^   =0,   ^  +  C  =  O, 

I  =   7H  =  f , 


L  /^  =  ±  i,  1-  =  ±  i- 


^  ;r  = 


3/(7  -  iKi  -  ^) 


(^+y-)^ 


Inverse  Transformations. 


\  A  =  C  =  -  B, 

I  —  m  —  f , 


XVI...  ^ 


A  =  ±  /i, 

A   =    ±    K. 


^   /v^-yV7^         V/^  -  v? 


V/-I-V/  '      y-V^-y  V?^ 


_y^v^-y         I -'t'"^ 


y  Vi  —  /— y  I  — Vi  —  ^ 


»  -^  =  -- 


I  —  Vi  —  ^'  j'  VI  —t  —J 


298  LINEAR  DIFFERENTIAL   EQUATIONS. 

(12)  Let  71  =  2,  7i'  =  4.     It  is  required  to  determine  two  poly- 
nomials R  and  5  such  that 

R'  -S'=  Ht\\  -t)\ 

where  H  is  a.  constant  introduced  for  convenience.  From  this  we 
deduce 

R  -  S'=  H'r,      R-\-  S'  =  N"{i  -^y. 

In  fact,  the  equations  7?— 5'  =  o,  R  -{-  S'  =  o  can  have  no  other 
roots  than  o  and  i  ;  besides,  these  equations  must  admit  no  com- 
mon roots,  because  a  common  root  would  annul  both  R  and  5.  One 
of  the  equations  admits  then  only  the  root  /  =  o,  and  the  other  only 
the  root  /=  i,  at  least  unless  the  first  member  of  one  of  them  re- 
duces to  a  constant,  a  case  which  will  be  examined  later.  5"  is 
compelled  to  be  of  the  first  degree,  but  R  may  be  of  the  first, 
second,  or  third  degree.     Suppose  R  to  be  of  the  first  degree, 

R  =  at  -\-  d,      S  =  mt  -j-  n, 
R-  S'  =  -  m'f  -\-{a  -  2mn)t  -^  b  -  7i\ 
7?  +  5'  =  m'f  +  («  +  2vni)t  +  3  +  7i\ 

As  we  can  always  suppose  ;/  :=  i,  we  ought  to  have  simultaneously 
(^  =  I,     rt  =  2ni,     nf  A;-  a  -\-  2in  -j-  2  =  o,     2;/^^  -\-  a  -{-  2i)i  =  o, 

which  is  impossible. 

Take  now  R  of  the  second  degree, 

R  =  af^  bt-^  c. 

If  neither  R  —  S"^  nor  R  -\-  S'^  reduces  to  the  first  degree,  we  shall 
have 

c  —  \,     b  =  2m,     a  -\-  ;«'  -|-  4m  -|-  2  =  o,     2a  -\-  2m''  -|-  ^m  =  o, 

giving  7n  =  —  i,  and  consequently  S  =  i  —  t. 

Suppose  R  —  vS^  reduces  to  the  first  degree  ;  we  must  then  have 

c  =  I,     a  —  ;//,     2?;z'  -\-  2m  -\-  2  ^  b  —  o,     4;//'  -\- b  -\-  2m  =  o,. 


Ek  GOURSAT:  HYPERGEOMETRIC  SERIES.  299* 

giving  ;;^  =  i,  <5  =  —  6,  «  =  i,  and  consequently  the  identity 
{f  -  6/  +  \y  4-  (/  4-  ly  =  -  16/(1  -  t)\ 
If  7?  is  of  the  third  degree,  we  must  have 

R  —  {mt  +  fif  =  t\     R-[-  {mt  +  fif  =  {i  —  t)\ 
from  which  we  derive 

[(I  -ty  -f]  =  2{mt^n)\ 
which  is  impossible  since  the  two  roots  of  the  equation 

(i  -/)=-/'=  o 

are  distinct.     If  one  of  the  functions  R  —  6"^  i?  -|~  '^'^  reduces  to  a 
constant,  R  will  necessarily  be  of  the  second  degree.     Let 

R  =  m^f  -f-  2;«/  -\-  c,     5  =  mt  -\-  i, 

ie  +  S' =  2;/^Y  +  4;;^^  +  ^  +  I. 


We  must  have 


—  I,     in  =^  —  2, 


and  consequently  the  identity 

(4/^  -  4/  -  i)^  -  (2/  -  ly  =  16/(1  -  /); 

this  can  be  deduced  from  the  preceding  identity  by  changing  /  into 
/ 


The  following  is  a  table  of  the  corresponding  transformations : 
XVII...  J      l=h  ^/^=l,       - 


'A^B  =  o,  C  =  o,^ 
XVIII...  J      /=f,  m  =  ^, 


X 


-\6t{\-ty    ~  -i6/-'(i-0' 

(I +4/ -4^7 


16/(1  -  /)    ' 


A  =  ±i,  }x=±\. 


.=  i'  +  ^L..=   (^-')' 


X 


16/(1  -  ty       16/X1  -  ty 

2/  -  \y 
-  16/(1  -  /) ' 


:3oo 


XIX. 


LINEAR  DIFFERENTIAL  EQUATIONS, 
A  =  O,    C  =  O, 


/  =  i,  m  =  I, 


(/^-6/+i)'  (^'+4/-4r 

^  =  — 7 ; — 7n ,  ■*■  ^ 


XX...  A 


(    A=o,  C=o,     1 
1=  m=.  I, 


(I +4/ -40' 

2/  -   I)* 


(2-/)^ 


(2  -  ty 


^-(/^-6/+i)-  ^-(/^+4^_4f 

(2/  -  ly 


XXI. 


A  =  ±  i,  r  =  ±  i.  J 

[A  =o,  B+C=^o,^ 
1=1,  m  =  h 


X  = 


X  = 


(1+4^-40^' 

i6/(i  -  ty         i6t\i  - 1) 


XXII...  \ 


A  =o,  B^C=  o, 
l=m  =  l,        \ 


—  i6/(i  —  t) 

'  {2t  -  ly 


4  }      -^    


{2-ty  ' 


-  i6(i  -  /)'        _  -  i6/Xi  -  /) 

i6/(i  —  /) 
^-(1+4^-40'' 


r  ^  =  (r,  B  =  2A, 


Inverse  Transformations, 
{x'-6x  +  I)' 


XXIII. 


l6;tr(l  -;iry  ~  ^'    l6^(l-  ;ir)'  ~  ^' 


^     (i  +  ^)'     ~     {x'-6x-\- ly  ~  ' 


XXIV. 


.=±f=±.. 


/  =  I,  ;;/  =  I, 


l6;ir(l    —   ;iry  _        —l6x{l—xy_ 


I   (I  +  ^y 

(;f'+  4;ir  —  4) 


=  / 


{x'-6x-\-iy 
(2  -  xy 


i6x\i—x)         '   i6x\i  —  x) 


=  i, 


i 


A. 

L  2 


±     yU    =     ±     r. 


(^r^+4-r  -  4)° 
(2  -  ;r)^ 


=  /, 


(2  -  xy 

(;ir'4-4Jt'-4)= 


=  /, 


l6;r'  (l   —  ;r)  _        —l6x^{\—x) 


(2  -  ;r)^ 


(A'=+4;f-4y 


GO  UK  SAT:  HYPERGEOMETRIC  SERIES. 


301 


XXV... 


/  =  I,    VI  =  I, 


f  (i+4^-4^T  _  ^     {2x  -  ly    _ 


^^  =  ±  >w  =  ±  -• 


16^(1  —  x) 

(1+4^-4^7 


—  i6x{i—x) 


_      {2x  -  ly 

{2x  -  ly    ~^'  {i-\-4x-4xy  ~  ^' 

—  l6x(l  —  ,r)  l6x(l   —  ;ir) 


I  (2^-ir 


(1+4^-4^7 


(13)  Let  71  =  2,  n'  =  T,.     R  and  5  are  to  be  determined  in  such  a 
way  that  we  shall  have 

R'  -  S'  =  Ht\\  -ty. 

R  can  be  of  the  first,  second,  or  third  degree,  and  S  of  the  first  or 
second  degree,  giving  in  all  six  cases  to  examine. 

First  Case. — Suppose  R  and  vS  each  of  the  first  degree, 

R  z^  at  -\-  b,       S  =  mt  -\-  n. 

The  polynomial  R'^  —  S^  will  be  of  the  third  degree,  and  will  admit 
one  double  factor  and  one  simple  factor.  Let  t  be  the  double 
factor ;  we  must  then  have 

b""  =  7i\      2ab  =  ^J7in\      (a-j-by  —  (;;z  -|-  ?^)'- 
We  can  always  take  ^  =  i,  ;/  =  i.     On  doing  this  we  have  a  =  ^— 


and  consequently 


I 


(f +,)•=(,„+.).; 


this  gives  ;;«  =  —  f,  a  =^  —  ^,  and  therefore  the  identity 
(9/  -  8)^  -  (4  -  3^)'  =  -  27t\i  -  ty 


Second  Case. — Suppose  R  of  the  second  and  S  of  the  first  degree^ 


VIZ., 


If  we  had 


R  =1  at^  -\-  bt  -\-  c,      S  =  mt-\-n. 
R'  -S'=  Ht\\  -t)\ 


302  LINEAR  DIFFERENTIAL  EQUATIONS. 

we  should  have 

5'  =  i?'  -  Ht\i  -  t)\ 

Now  the  equation  R'^  —  Hf{i  —  /)'  =  o  breaks  up  into  two  equa- 
tions each  of  which  is  at  most  of  the  second  degree,  viz., 

R=  ±  VIT.  t{i  -  t), 

and  consequently  cannot  admit  of  a  triple  root ;  S^  —  R""  will  then 
liave  a  triple  factor  and  a  simple  factor.  If  /  be  the  triple  factor,  we 
shall  have  the  conditions 

■c^  =  7i\      2bc  =  ynn\     b"  -\-  2ac  =  yn^n,     {a  -\-  b  -\-  cf  ^^  {in  -\-  lif. 

Taking  «  =  i,  ^  =  i,  we  get 

yn  yn^       gm'        gnv"  ^ 

and  so 

w  =  -  |,     b  =  -  ^,     a  =  ^\, 

from  which  results  the  identity 

(8/'  -  36/  +  277  -  (9  -  80'  =  -  64^^(1  -  /). 

Third  Case. — Let  R  be  of  the  third  and  5  of  the  first  degree, 

R  =z  af  ^  bf  +  r/  -f  ^/,     5  =  int  +  n. 

R"^—  5'  will  admit  of  a  quadruple  factor  and  a  double  factor,  or  two 
triple  factors,  or  a  quintuple  factor  and  a  simple  factor.  We  cannot 
have  R'  —  S'  =  t\\  -  t)\  for  then 

S'  =  R'  -  t\i  -  t)\ 
In  order  that 

R'  -  t\i  -  ty  =0 

should  admit  only  one  triple  root,  we  ought  to  have 

R  =  f{i  -  t)  ^  H', 


GOURSAT:   HYPERGEOMETRIC  SERIES.  303 

■but  then  the  equation 

R  +  t\i  —  t)  =0 

becomes  it^i  —  f)  -{-  H  —  o  and  has  no  triple  root.     Neither  can 
we  have  R"-  -  S'  =  Ht\i  —  t)\  for  then 

R'=  S'  +  Ht\i  -ty, 

and  in  no  case  can  the  second  member  of  this  admit  three  double 
factors.     It  remains  to  be  seen  whether  we  can  have 

R'  -  S'  =  Ht\\  -  t). 

Taking  n  ■=■  i,  d  ^=  i,  we  must  have 

2c  =  yn,      2c^  -j-  4(5-  =  Gi/i",     2bc  -]-  2a  =  m^,     b"  -\-  ac  ^=  o, 


givmg 


3;;z  3;/z  iff 


and  substituting  in  the  fourth  relation  this  becomes 
-^  -  ^^  =  o,  givmg  m  =  o. 

This  hypothesis  must  then  be  rejected. 

Fourth  Case. — Suppose  R  of  the  first  and  5  of  the  second  degree, 

R  =  at  -\-  h,      S  =  inf  +  nt  +  p. 

R^  —  S^  cannot  admit  two  triple  factors  ;  if  we  had 

R'  -  S'=  t\i  -ty, 
then 

R'=  s'-{-t\i  -  ty, 


304  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  the  second  member  evidently  admits  more  than  one  distinct 
factor.  If  we  had  S^  =  R""  —  t\\  —  t)\  each  of  the  two  equations 
/l  =  ±  f{l  —  /)  should  admit  a  triple  root : 

R  -  t\i  -  t)  =  {iit  +  v)\ 
Then 

2R  =  {ut  +  vY  +  (///  +  tO'; 

R  being  of  the  first  degree,  this  is  impossible.     If  we  had 

S'  -  R'  =  Ht\i  -  t\ 

we  should  find  as  condition  ;/  =  o,  ab  =  o.  This  hypothesis  there- 
fore gives  nothing. 

FiftJi  Case. — R  and  5  of  the  second  degree.    We  can  demonstrate 
as  in  the  preceding  cases  that  this  hypothesis  is  to  be  rejected. 

Sixth  Case. — R  is  of  the  third  and  5  of  the  second  degree, 
R  =  af  ^  bf  -\-  ct  ^  d,       S  =  mf  +  nt  -f  /. 

R^  —  vS'  may  be  of  a  degree  lower  than  the  sixth.  By  simple  trans- 
formations the  second  member  of  the  identity 

R-"  -  S'  =  HV\\  -  ty 
is  conducted  to  one  of  the  following  forms: 

Hr,      Ht\      Ht,      Ht\i-t)\      Hf{i-t),      Ht{i-ty 
If,  for  example,  we  had 

R'  -  S'  =  Ht\i  -  t), 

changing  /  into  —  and  multiplying  by  f,  we  deduce 

R^'  -  S,'  =  -  Ht{\  -  t).. 
If  we  had 

R'  -  S'  =Hl\ 


I 


GOURSAT:  HYPERGEOMETRIC  SERIES.  305 

we  should  conclude  that  the  equation 

R-^  _  Hf  =  S' 

had  three  double  roots,  which  is  impossible.     Supposing  we  have 

R'  -  S'  =  Ht\ 
then 

R'  -  Hf  =  S\ 

and  we  should  have 

R  -  X^IT.  t  =  {lit  +  v)\ 

R  +  viT.t  =  {ti,t  +  v:)\ 

and  consequently 

2  ^li.  t  =  {lit  +  vy  -  {u,t  +  v^\ 

which  is  inadmissible.     If  we  had 

R'  -  S'  ^  Ht\\  -  t)\ 
then  we  should  have 

R  +  S^IT .  t{\  -  t)  =  {lit  +  v)\ 
R  -  VIT.  t{i  -  t)  =  {uj  +  v,)\ 
and  consequently 

{lit  +  vY  -  {11, t  +  v,Y  =  2  V7/.  t{i  -  t). 

That  this  may  be  so  we  must  have 

ut  -\-v  =  t  -\-j\      uj  -\-  v^  —  t  +y'; 
from  which 

R  =  w  +7T  +  w  +rr  =  i'  -  r  -  f  +  ^ 
s  =  {t+j){t+r)  =  f  -t  +  i, 

and  consequently  the  identity 

{2t^  -  3t^  -  3/  +  2y  -  4{f  -t+iy=-  27t\i  -  t)\ 


3o6  LINEAR   DIFFERENTIAL  EQUATIONS. 

If  we  wished  to  have 

R-"  -  S'  =  Ht\\  -  t\ 
we  should,  in  supposing  d  =■  p  =^  i,  get  the  conditions 

nt"  =  «*,      ^m^n  :=  2ad,     3;;z*  -\-  ^m?i''  =  d'^  -\-  2ac,     3«  =  2^, 

(;;,   +   ;^   +    1)3   =    (^   _^   ^   +   ^  _|.    j)^. 

these  give 

3  4/^  3« 

a  ^  m  Vm,      0  =  ,       ^  ^  — ; 

2  2 

substituting  these  in  the  third  relation,  we  have  m{2  Vm  —  2f  =  o. 
If  we  take  n  ■=  2  Vm,  we  get 

a  ^^  m  Vm,      b  :=  im,      c  =  ^  Vm, 
and  are  so  led  to  the  identity 

{mt  -\-  2  Vm  t  -\-  i)'  —  {m  Vm  f  +  ynf  -\-  3  Vm  /  -f  i)"  =  o 
this  hypothesis  gives  therefore  no  transformation.     Suppose 

R-"  -  S'^  Ht{\  -t); 
the  conditions  now  are 

m°  =  «^      ■^in'^n  =  2ab,      yn'^  -\-  ymi^  =1  b"^  -\-  2ac, 
6mn  -\-  n^  ■=  2a-{-  2bc,      yn  -j-  yi^  -j-  yi  ^  c^  -\-  2b  -\-  2c ; 
these  give 

. —  3«  Vm  6mn  -\-  n^  —  2111  Vm 

a  -—  m  Vm,      b  = ,      c  =  -r= ; 

2  yt  r ;« 

substituting  these  in  the  third  equation  gives 

n^  —  \2mn  -\-  i6m  Vm  ■=■  o, 

or,  making  n  =  u  Vm, 

u^  —  12U  -{-  16  =  {u  -\-  4)(u  —  2)'  =  o. 


GOURSAT :  HYPERGEOMETRIC  SERIES. 


1P7 


AVe  have  therefore  either  n  ^=  2  Vm,  or  «  =  —  4  Vm.  If  we 
take  ;^  =  2  Vm,  we  come  to  the  same  identity  as  above.  Take 
n  =^  —  4  Vm ;  now 

a  =  m  Vm,      d  =  —  6;n,      c  =  ^  Vm, 
Substituting  these  in  the  last  relation,  we  find  Vm  =  4;  therefore 

ju  z=  16,      n  =  —  16,      «  —  64,      d  =  —  96,      c  =  30, 
giving  tjie  identity 

(64/=  -  96/'  +  30/  +  I)'  -  (i6/»  —  16/  +!)=■=-  108/(1  -  t). 
,   U  R''  —  S'  =  Ht,  the  conditions  are 

cC  =  m^,      yj^7i  =  2ab,      yii^  -\-  ynn^  =  (5'  -|-  2ac, 
6mn  -\-  n^  =  2a  -\-  2bc,      c^  -\-  2d  =:  yn  A-  ^n^, 


giving 


15  V~i 


ti  =^  —  4  Vm,      a  =  m  Vm,      d  =  —  6m,      c  =   ^^  ^  ^ 

2 

these  substituted  in  the  last  relation  give  m  =  o. 

The  following  is  a  table  of  the  transformations  deduced  from  the 
preceding  identities: 

A-^B  =  o,     C  =  o,     l  =  h     Jn  =  h     ^  =  ±h    1^  =  ±  h 


XXVI..  J 


{9t  -  8)' 


(i-gty 


,  X  — 


_{9-mt 

2'jf{\  —  ty^     —2'jt{i  —  tf'^     27(1—/)' 


27/ 

(/  +  8)-(i-/) 
—  27/^ 


27(1-0^ 


3o8 


LINEAR  DIFFERENTIAL  EQUATIONS, 


XXVII... 


64^' ( I  —  t) 


—  64/(1  —ty 


_  (8  —  36/  -\-  2yt'y        _  {2jf  -  18/  -  i)' 


64(1  -t) 


64/ 


_  (/'+  18/  -  27)^        _  {f  -  20/  -  8)' 


XXVIII x  = 


64/'  '  64(1  -/)'     ' 

(2/3  _  3/'  -3/4-  2)' 
-27/^(1-"/)^       ' 

(64/'-96^y-30/+iy       _  (/°H-30^' -96^+^4)' 
^'lo8/  (I  -  /)       '  "^  ~         loSr  (I  ^  /) 

(/^- 33^° -33^+ I)' 

108/(1 -/y 

^  -f  ^  =  o,     C  —  o,     /  =  I,     ;;2  =  i,     A  =  ±  i,    /^  =  ±  i- 


XXIX...  ^ 


Jtr  = 


XXX... 


(3^  -  47        ,__(3L+iL    ,.       (3  -  4^.y 
•^-_  27^(1-/)'  27/(1 -/y"^- 27(1  -/.)'' 

_{4^-il    ^_(A-^     ,.   ..    (^  +  37 
■^  -      27/      '  -^  -     27/'     '  -^  -  27(1  -  // ' 


XXXI.. 


(8/  -  9)^ 


_   (8/  +  ly         _  (8  -  9/)V 


X 


-  64/^  (i  -  /)'  -^  ~  64/  (i  -  /)' '  64 (i  -  /)  '■ 

(9/  -  iy(i  -t)     _  (/  -  9)^(1  - 1) 


XXXII ;r 


X  = 


XXXIII...  ^ 


64/         '"^^        64/' 

_(8_+^_ 
-64(1-0'' 

4(/'  -  /  +  0° 

27/^(1  -  ty  ' 

( 1 6/'  -  16/  +  i)"     ^  _  (/'  -  16/  +  1.6)" 
108/(1  —  /) 

(I  +  14/  +  /T 


,   X  — 


-  108/^1  -/)  '■ 

I    _  ^i  -t-  14/  +  ^T 
[^~  -  108/(1  -  /y 

^  =  o,     (7  =  0,     /  =  i,     w/  =  f ,     A  =  ±  i,     r  =  ±  ^L 


I 


GO  URSA  T:   HYPERGEOMETRIC  SERIES. 


309 


,        (9^  -  8)'       ^.       (I  -  9ir        _(9-8/)V 


XXXIV...  ^ 


-(3/-4y'  (3^+0= 


(4^-3)^' 


{t-gn  (/  +  8)'(i-/)       _(i+8/)^(i-/) 


(^+3r 


(4-0^ 


(1-40'      ' 


_  (8/^  -  36/  4-  27)'        _  (8/^  +  20/  -  I)' 
"""         (9-8/)'        '  -^  -        (8/  +  I)' 


XXXV  !^-   (8-36^  +  270'       (27/-  - 18^  -  ir 


(/^  4-  18/  -  27)"  (/^  —  20t  -  8)' 

-f  = -.- T-  ,  X  = 


XXXVI. 


XXXVII...  ^ 


{g-t)\l-t)    '   "-  /(8  +  /y 

{2f  -   ^f  -  3/  +  2)' 
4(^^  -  /  +  ly        ' 

(64/^-96/'H-30/+i)'        _  (/34_3o^^_96/+64)" 
(^'-33^^-33^+ir 


^  -    (16/^  -  i6t  +  i)^    '  -^  ~      {f-  16/4-16/     ' 


-       (14-14/  +  /7       • 
A=o,     C=o,     /  =  I,     w  =  f ,     A  =  ±  |,     V  —  ±  ^. 


XXXVIII...  ^ 


(9/  -  8/ 


\a J  >    -^ 


(1-90''  (9 -80V' 

(4-/r  „  (1-40' 


;tr  = 


(/-9)V'  -^     (^4-8/(1  -0'''-(i4-80Xi-0' 

(9-80'  „_        (8^+0' 


;,  X 


XXXIX...  \ 


,  X 


(8/*  4-  2ot  -  if ' 
(1-90^1-0 


XL X  = 


(8/^-36/4-27) 

_        (9/  -  8)V 

~  (8  -  36/  +  27/^) '  "  ~  (27/^-  18/-!)" 

^  (9- /)-(!-/)      ^^_       (8  4-0'^ 
(/'  +18/-  27)^ '  (/^  -  20/  -  8)' ' 

4(/'-/4-iy 


(2/'  -  3/'  -  3^  +  2)' 


3IO 


LINEAR  DIFFERENTIAL   EQUATIONS. 


XLI. 


X  = 


{i6f  -  i6t  +  i)" 


X  = 


{t'  -  \6t  +  i6y 


(64/='-96/'+30^4-i)"  (/='+3o/' -96^+64)^ '• 

(I  +  w  +  ty 


A  =  o,     B  -\-C  =  o,     /  =  I,     m  =  ^,     fx=  ±i,     r  =  ± 


XLII...  i 


_  —  27/' (i  —  /)       _  27/(1— /)       _27(i-^_) 
"""       (3^-4)=      '  ''"  (3^+ir   '  ""-(3-40'' 


X  = 


27/ 


(4/  -  I)' 


X  = 


27/' 


(4  -  ^y 


X  = 


27(1  -/)' 

(^  +  3)' 


_  64/' ( I  —  /)  _  64/ (i  —  /)'        _  64 (i  —  /) 

■^  ~    (9  -  8/)^  '       "^^    (8/  +  I)'  '  "^"(S^:^/)'/" 


XLIII...  ^  ;ir  = 


64/ 


,  -^^ 


64/' 


XLIV ;ir  = 


(9/-  1)^(1-/)'  --(/-9y(i  -/)' 

-64(1-  ^T 

(8  +  /)V     ' 
27/^(1  -/y 


4(/^ -/+!)' 


XLV. 


'     —       108/(1  —  t)  _  —  108/^(1  —  /) 

^  -  (16/^  -  16/  +  i)= '  ^  ~  (/»_  16/ +16/' 

_  —  108/(1  -  ty 


A  =  o,     B  -{-  C  =  o,     /=f,     m  =  I,     lA  =  ±^,     r  =  ± 


—    4-1 


'        _  —  27/^(1    —  /)  _   —   27/(1    -  /)'  27(1   —  /) 


XLVI...  ^ 


(9/  -  8)> 

27/ 


,  -^ 


;tr  ^ 


(1-9/r 

(I  +  8/)^ (I  -  tV  ^  ~   {t-9Yt   ' 

—  27/' 
(/  + 8/(1-0' 


(9-8/)'/' 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


311 


-64/^(1  -t)        _   -64/(1  -ty 

(8/^  -  36/  +  27)^ '  ^  -  (8/^  +  20/  -  ly ' 


XLVII...  J  ;ir 


64(1  -/) 


;r  := 


,  x  = 


XLVIII X  = 


XLIX...  i 


X 


(8  -  36/  +  27/')= 

64/^ 
(/^  +  18/  -  27)' 

-27/^(1  -/)^ 

—  108/(1  —  t) 

(64/^-96/^+30/+!/ 
108/(1  — /y 


X  = 


64/ 


(27/^  -   18/  -   1/ 

64(1  "  0^ 
(/"  -  20/  -  8)'  ' 


,  x  = 


108/' (l  —  /) 


(/3  +  3o/»_96/  +  64)^' 


(^'-33^'^-33^+iy* 

Inverse  Transformations 
{  A=o,  4^+3C=o, 
/  =  f ,  ;;/  =  f , 


L...^ 


^  =  ±  i, 
X 

2 


(gx-sy  (3-^-4)°    _ 

27^Xi-,r)      '-27;i:Xi--^)~^' 


±    yU. 


(9-^  -  8)-  _     (^-  3£r  _  , 
^  (4  -  3-^)'  ~   '  (9^  -  8)^  -  ^' 
-27x\i-x)        -27x\i-x) 
j       (3-^-4)'    ~'     (9;.-87~-^- 


LI. 


/  =  f ,  m  =  f , 
^'  =  ±  h 

2 

C=o,  4^+3^  =  0, 

/  =  l,   ;//  =   5^ 


(I  -  gxy 


(3^+iy 


LII...  ^ 


^=  ±h 

V 

r  =  ±  y". 


-27^(1  -^)«      ^'  27^(1'-^)"  -  ^' 

(I   -  gxy  _       (3;t:+  i)=  _ 

(3^  +  ir~   '(I  -^~  ^' 
27^(1  —  ^)^  _      -27.r(i-,r)'  _ 

73^+7r"~  '  (1-9^-^- 

{g^^xyx  ^    (3  -  4;^)3  _ 

27(1-^)  '27(1  -;ir)  -  ^' 

(9  -  d>xyx  (4x  -  3)^    _ 

(4^-  3r  "^'(9^8^""^' 

27(1  --y)  _        27(1  -  ;r)  _ 
J  (3  -4^y  -^'   (9-8^)^^-^- 


312 


LINEAR  DIFFERENTIAL   EQUATIONS. 


LIII. 


-2'j{\-xy  ~  '  27(1  -  xy  ~  ^' 


'    —    2>    ^^^    —    3^> 


(^+3/    -''    {x-c^Yx-'^ 

27(1  -  ^y  _     -27(1-.^-)'  _ 


(^  +  3)' 


(^  -  9)V 


'  (:  =  4^,  i5+5^=o,l  (;r+8)'(i-.r)_       (4  -xy  _ 

-  27X-'        ~  ^'        27^"     ~  ^' 


L1V..J 


/  =  |,    ;;/  =z  1, 


X 

-  =  ±  y. 


{x+^)\i-x)_       (4-^r 


(4-^r  ^'(^+8)Xi-^)- 


27;ir' 


=  t, 


2'JX^ 


LV... 


'y^ 

=  4C  B^SC  =  o,^ 

I  =  I  m  =  h 

M  =  ±  h 

V 

A  =  ±  -. 
2 

{4-xy-"  {xJ^8y{i-x) 
{i-^sxy{i-x)_(4x-iy 


=  t. 


2'JX 


t,-- 


2'JX 


{i^%xy{i-x)        (i-4^r   _, 


(i  -4;ir)''    ~  '(l+8x)Xi-^-)" 


27:1: 


=  ^, 


27^ 


'^=0,  ()B^%C^o, 


LVI. 


/  =  1,    ;;/  -   5 


^' 


¥» 


=   ±yu. 


(8^^  -  36^  +  27)' 


=  /. 


-  6\x\  I   —  ;ir) 
(8^  -  9) 


A 


—  6^x\i  —  x) 
(8;^'  -  36;tr  +  27) 
(9  -  8-^)" 
(9  -  8^')' 


=  /, 


=  ^, 


(8^-^  -  36^  +  27) 
—  64^'(i  —  x^ 


i  =  t, 


=  t. 


{Sx  -  gy 
—  64x\]  —x) 
iSx'  -  36X  +  27)^ 


=  /. 


GO  UK  SAT:  HYPERGEOMETRIC  SERIES. 
'  {^Zx"  +  20;f  -  I) 


313 


A  =0,  B  =ZC, 


/  =  f ,  in  =  i, 


LVII... 


A=  ± 


3' 


—  64x{i  —  xy 

(8-^  +  iy  ^, 

64;r(l  -  Af 
(8.r^  +  20;tr  -  i)' 

(87+77 

(8-^  +  I)^ 

(8^"  +  20;ir  —  ly 

64r(i  —  x)* 

—  64;ir(l  —  xy 
{Sx'  +  20;ir  —  I)' 


=  ^, 

=  A 

=  ^, 

=  t. 


5 
"6"' 


LVIIL 


(7  =  0,  9^+8^=0, 
/  =  f ,  m 

V 


f  (I  -  36,v  +  27;.^  _ 

64(l-;t:)  ~     ' 

(8  -  gxyx  _ 
64(1  -  ^)   ~    ' 

(8  -  36^  +  27;^')' 
{gx  —  syx 
{gx  -  Sfx 


(8  -  36X  +  27;.:")' 

64(1  —  x) 

(8  -  gxyx  ^  ^' 

64(  I  —  ;ir) 


LIX... 


(7: 

=  0, 

B 

= 

8^, 

/ 

=   h 

m 

1 
3 

i, 

X  = 

--  ± 

^ 

V    = 

=  ± 

3' 

(8  -  i6x  +  27;r^) 
(8  +  20;r  —  ^r')' 

64(1  -  xy 

-  (8  +  ;r)';ir 


=  t, 
=  t. 


64(1  —  x) 

(8  +  20;r  -  ;i;') 
(8  +  xyx 
(8  +  xyx 

(8  +  20;ir  —  ;ir^) 

64(1  -  xy 


=  t, 
=  t. 


=  t. 


-(8  +  xyx 
64(1  -  xy 

L  (8  +  20;r  —  x'y 


=  t. 


314 


LINEAR  DIFFERENTIAL  EQUATIONS. 
'  {27 X"—  \%x  -   I) 


A  =  ^C,  B^ioC^o, 


I  =  I  m  =  I 


LX..J 


LXI. 


LXII. 


V 

X=  ±- 


64X 
(gx  -  i)''(i  —  ^-) 

64X 
(2yx'  —  i8x  —  i) 
(i  -9xy{i  -^ 

(I  -  9x)\i  -  x) 
(27A''  —  i^x  —  i) 

Jgx  -  1)^(1  -  x) 
64X 

.  {2'/x'  —    l?,X  —    I) 

(-y'+  \%x  -  27)^ 


-  =  t, 

=  t. 


fC=9^,^+ 10^  =  0, 

/  =  1,    ;//  =  I, 


\ 

3 


64;!:' 

(;,   -   9)Xl    -   X) 
64X' 

(;tr'H-  18^'  -  27) 
(9  -  ^f  (T^^ 
(9  -  ^)Xl  -  ^') 


=  ^, 


(;t-'+  i8;ir  -  27) 
64;ir' 


=  t, 
=  t, 


(^  -  9)Xi  -  ^) 
64;ir^ 


A  =  C=  -  B, 
I  =1  in  =^  t. 


X  =   ±  fX  =   ±   y. 


{x'  4-  i8;ir-  27) 

{2x'  -  3.y'  -  3^  +  2)' 
—  27;i-'(l  —xY 

2yx\i  -  xf    ~  ^' 

(2^'  _   3^.=  -   3;r  +  2)' 


4(-^^  -  ^  +  I)' 

4(;P»    -    ^   +    I)  = 


{2x'  -  3^'  -  3-^  +  2)' 

2yx\i  -  xy     _ 
4{x'-x~^  1)3  -^ 

-  27;trXl  —  xy 
{2x^  -  3^'  -"^~+~^ 


=  ^, 


=  / 


=  /> 


=  /. 


GOURSAT:  HYPERGEOMETRIC  SERIES.  3 1 5;, 


LXiXI.  .  \ 


A  =  t  >u  =  ± 


LXIV. 


{  B+C^o,  C^~  i6A, 


-  =  ±  yu  =  ±  ^. 
4 


—  io8;r(i  —  x) 
(l6;r^  -   l6;tr  +  I^  _ 

lo8;ir(l  —  x)         ~    ' 

(64;!:''  -  96;tr'  +  30;tr  -|-  i)^ 
1 6;ir'  -  l6;r  +  Ip 
(i6.r'  -  \6x  +  ly 

(64;!:='  -  96;ir^  4-  30;ir  +  1/ 

lo8;r(i  —  x) 
{i6x'  -~i6x  +  if  "^  ^' 

—  lo8;ir(l  —  x) 

^  {64X'  —  96^'  +  30^  -j-  l)' 

{x'  -f-  30;^"  —  96;^  -|-  64) 


=  /•, 


=  i, 


=  t. 


io8;ir'(i  —  x) 
{x^  —  i6x  -\~  i6y 


/, 


LXV...^ 


A  =  C,  B  =  14C, 
/=  f ,   m  =  ^, 


u 

X  =  ±-  =  ±  V 
4 


—  108^X1  —  ^) 
{x'  +  ^ox'  —  g6x  +  64)' 

(^''  —  \6x  ~\-  i6y 

jx'  -  i6.y  +  16)^ 

(;f^  +  30;tr  —  <^6x  -\-  64)' 

—  I08,t:Xl    —  ^)     _ 

(;r^  —  16^  +  16)^  ^  ^' 

loSx^i  —  x) 
{x'  +  30^-^  -  96;r  +  64)^ 


io8x(i  —  x)* 

(I  +  14-t-  +  xy  ^  ^ 

—  io8^-(i  —  xY  ' 

(.y^  -  33^^  -  33.r  +  i)^ 

(I  4-  i4A-  +  ;ry 

I  +  14X  -\-  x'y 

{x'  -  33.r-^  -  33,r  -f  1)' 

—  ioSx{i  —  xy  _ 

(I  +  Hx  +  xj  -  *' 
io8;r(l  —  xy 


=  t, 
=  t, 

—  t^ 


3i6 


LINEAR  DIFFERENTIAL   EQUATIONS. 


(14)  We  still  have  the  hypothesis  n  =  3,  n'  =  4  to  examine.  It  is 
easily  shown  that,  under  the  imposed  restrictions,  it  is  impossible  to 
have  an  identity 

R'  -  S'  =  Hr{i  -tj; 

there  exist  therefore  no  other  rational  transformations  than  those 
already  determined. 

(15)  In  combining  among  themselves  the  transformations  which 
we  can  effect  for  the  same  values  o{A,B,  C,l,  m,  we  obtain  new  alge- 
braic, but  not  rational,  transformations.  It  remains  to  demonstrate 
that  all  such  transformations  are  obtained  in  this  way. 

Resume  the  two  differential  equations  (5)  and  (7)  which  are  to 
be  satisfied  by  the  function  x  of  t. 


VAx'-{-  Bx-\-C  ,          VA'f  +  B't  +  C  _,, 
(5)  A^ ^^-  dx  = ;- ^ dt. 


X\l   —  X) 


(7) 


dx 


/(I  -  /) 
Kdt 


x\\  —  xf     f{i  —  ty 

The  relation  deduced  from  these  is 


(8) 
where 


{Ax'  +  Bx  +  cy~  {A'f  +  B't ^cy 


<f>{x)  =  X{X-  1){2AX  +  B)^2{Ax'-\-Bx  +  C)[{1-  l){x-  l)+(;;z_  l);r], 

<fyit)  =  f{l-i){2A't+B')-\-2{A'fJrB't-{-C'W-i){t-i)^{m'-i)t]. 
Differentiating  (8)  and  taking  account  of  (5),  we  get  the  new  relation 
x(x  -  i)[2cp'{x){Ax'^  Bx+O-  3cp(x){2Ax  +  B)] 

^9)  (Ax'-^Bx^  cy 

_  /(/  -  i)[20/(/)(^7'  +  ^V  +  n  -  l<t>kt){2A't  +  B')-\ 
~  {A'f^  B't^Cy 


GOURSAT:  HYPERGEOMETKIC  SERIES.  31/ 

Every  integral  common  to  (5)  and  (7)  satisfies  (8)  and  (9).  Con- 
versely, every  algebraic  function  which  satisfies  simultaneously  (8)1 
and  (9)  is  a  common  integral  to  (5)  and  (7).  In  fact,  every  function, 
which  satisfies  (8)  satisfies  also 

\/ix  — j—  ijX  — |-  C  ) 

-    ^A't^-^^-B'T:^  CJ VAt  +  Bt+C   .dt,. 

or,  taking  (9)  into  account,  equation  (5).  Taking  account  of  (5),  (8) 
can  be  written 

^.  log  [  VAx'  -\-  Bx  ^  C  .  x^-\i  —  x)'"-'']  . 

=  ^  .  log  [  \/A'f-\-B'^-\-C'  .  /^'  -  '(i  -  /)'«'  -  ^]. 
We  have  then 


VAx'-{-Bx-\-C.x^-'{i-xy"-^=  \/A'f-\-B't+C'.t^'-\i-ty"'-\.. 
and  consequently,  still  taking  account  of  (5), 

dx  Kdt 

x\\  —  x)'"  ~  f{\  —  tY" 

We  are  thus  reduced  to  the  problem  of  finding  the  conditions  to  be 
satisfied  in  order  that  equations  (8)  and  (9)  shall  have  one  or  more 
common  roots. 

(16)  To  this  end  we  shall  first  demonstrate  the  following  theo- 
rem : 

Whenever  (8)  and  (9)  have  a  common  factor  of  degree  higher  than 
the  second  with  respect  to  one  of  the  variables,  they  are  identical. 

Suppose,  for  example,  that  they  have  a  common  factor  of  degree 
n  in  X  (n  ^  3).  Let  x  and  x^  be  two  values  of  x  corresponding  to 
the  same  value  of  /.  Between  x  and  ,r,  there  will  be  a  symmetric 
relation  of  degree  n,  at  least,  which  should  be  contained  in  each  of: 
the  following : 


(10) 


{Ax'  -\-Bx-{-  cy  ~  {Ax:+  Bx,  +  cy 


31 8  LINEAR   DIFFERENTIAL   EQUATIONS. 

x{x  -  i)[2^\x){Ax'-\-  Bx+C)-  3cp{x){2Ax  +  B)] 


.(II) 


{Ax'  -\-  Bx-\-  cy 

xjx-  i)l2ct>{x,){Ax:  +  Bx,  +  C)-  30(^.)(2^.f.  +  m 
{Ax:^Bx,^Cy 


Multiply  the  first  of  these  by  3  and  add: 

(12)  60(;ir)[(/—  \){x  —  \)^{in—  i)x']  +  2x{x—  i)(p' (x) 

'  {Ax'  +  Bx-\-  cy 

=  60  (^,)  [(/  -  i)(-^.  -i)-^{m-  !>.]  +  2x,  {x-  i)0X-y.) 
{Ax:  +  Bx,  +  Cy 

We  can  replace  the  system  (10)  and  (11)  by  (10)  and  (12).  For 
brevity  write 

iix)  =  6cp{x)[{l-  i)(;ir  -  l)  +  (w  -  l)^]  +  2x{x  -  l)cp' {x). 

If  n  =  5  or  6,  the  relation  (12),  which  is  at  most  of  the  fourth  de- 
gree, must  be  an  identity,  and  consequently  equations  (10)  and  (11) 
become  the  same.  Let  ?i  =  4.  If  (12)  is  not  an  identity,  it  must 
be  an  equation  of  the  fourth  degree  ;  this  requires 

A^o,    A-\-B+C^o,     C^o. 

All  the  roots  of  (12)  should  belong  to  (10);  this  requires  that 
Ax*  4"  ^^  -\-  C  he  a.  perfect  square.  In  fact,  let  ^  =  «  be  a  simple 
root  of  y^;f'' -|- -^-^  +  C  =  0.  Consider  the  two  equations  (10)  and  (12). 
For  x^  =  a  two  values  of  x  become  equal  to  a  in  (12),  and,  from 
the  form  of  this  equation,  we  have  for  these  values 

Hm  l-^^^]'  =  I. 


To  prove  this  it  sufifices  to  write  x  =  a  -}-  s,  x^  =  a  -\-  z^;  equation 
(12)  can  then  be  written 

I  +  e       I  +  e, 


P-  GOURSAT:   HYPERGEOMETRIC  SERIES.  319 

where  e  and  ej  are  infinitely  small  at  the  same  time  as  z  and  ^, ; 
from  this  we  evidently  deduce 


In  like  manner,  in  equation  (10),  for  ^,  =  or  three  values  of  x  become 
equal  to  a,  and  we  have  for  these  values 


lim  I \   =  I. 

.X,  —  a 


The  two  values  of  x  which  satisfy  the  first  condition  can  evidently 
not  satisfy  the  second.     It  is  necessary,  then,  that  we  have 

Ax''  -^  Bx  ^  C  =  A{x  —  of  ; 
consequently 

<l){x)  =  2A  {x  —  a)\x  {x  —  \)  -\-  {x  —  a)[{l  —  l){x  —  i)  -\-  [in—  l)x'\\. 

Relation  (10)  reduces  to  the  fourth  degree: 

^A\x{x-i)^{x-a) \{l  -  i)f^  -  I)  +  {,n  -  I);.]  Y 
^     '  A\x-ay 

^  4A  \ x,{x,  -  I)  +  {x,  -  a)l{/  -  i){x,  -i)-Jr{m-  i)x,']  ^ 

A^{x,-ay  —  • 

In  like  manner  (12)  can  be  written 

.     .     4Ax^  (^- !)'+  {^-^)A  (-y)  _  4Ax:  (x,  -  ly  4-  (x-  oc)/^  [x,) 
^^  A\x-ay  -  A\x,-af  • 

Relations  (10)  and  (12)    should  be   the   same.     Subtracting   them, 
member  from  member,  we  get  a  new  relation. 


A\x-aY~  A\x^-ay' 
•which  should  be  an  identity,  as  it  is  at  most  of  the  third  degree. 


320  LINEAR  DIFFERENTIAL  EQUATIONS'. 

Suppose  ??  =  3  ;  if  (12)  is  not  an  identity  it  should  be,  at  least,, 
of  the  third  degree,  since  two  of  the  quantities  A,  A  A^  B  -\-  C,  C 
cannot  vanish  at  the  same  time.     Let 

A%o,    A-\-B^C%o. 

There  are  several  cases  to  be  examined. 

First  Case. — Ax^  -f-  Ex  -|-  (7  =  O  admits  two  simple  roots  x  ^  a, 
;ir  = /5  different  from  zero.  cp{a)(l){(f)^o.  We  cannot  have  simul- 
taneously ip{a)  =  O  and  ?/'(/5)  =  o,  for  then  (12)  would  be  of  a  degree 
lower  than  the  third.  Suppose  y.' («)?/' (/?)  <  o;  for  x^  =  a,  two  values 
of  X  become  equal  to  a  and  two  to  /?,  by  (12) ;  let  us  assume  that 
the  two  values  o£  x  which  become  equal  to  a  belong  to  (10).  From 
(12)  we  should  have 

lim  ( )   =  I, 

\x^  —  aj 

and  from  (10) 


lim  (- -)   =  I, 

\x^  —  aJ 

two    incompatible    conditions.       We    reason    in    the    same    way    if 
tp{^)  =  o.     This  hypothesis  must  therefore  be  rejected. 

Second  Case. — Ax''  -\-  Bx -\-  C  =  o  admits  the  simple  root  x  =  o 
and  another  root  x  =  at  different  from  unity.  Equations  (10)  and 
(12)  become 


(10) 


(12) 


[<P{x)J     _     [0(^,)]' 


x{x  —  ay      x,(x^—  «)"' 
x{x  —  aJ      x^  {x,  —  ay  ' 


f{x)  is  of  the  third  degree,  and  we  should  have  ip  (a)  <o,  0  (a)  ^  O. 
Reasoning  as  before,  we  find  that  this  hypothesis  must  also  be 
rejected.  .  ■ 

TAtrd  Case.— Ax'  ~{- Bx -\-  C  =  A{x  —  a-)'.     Replace  the  system  " 
(10)   and   (12)   by  (10)  and   (13).     This   last  equation   must  be  an 


I 


GOURSAT:  HYPERGEOMETRIC  SERIES.  321 

identity,  otherwise  it  would  be  of  the  third  degree,  and  we  should 
derive  from  it 

Hm  r  ~  ^\   =  I, 
\;r,  —  aj 

while  (12)  gives 

T       /x  —  a\ 

lim    — =  I. 


I  \^,  —  a 

Summing  up  :  When  n  is  equal  to  or  greater  than  3,  one  of  the  equa- 
tions (12)  or  (13)  reduces  to  an  identity  which  requires  that  one  of 
the  rational  fractions 


{Ax'  -\-Bx-^  Cy '     A'{x  -  ay 

reduces  to  zero  or  a  constant.  It  is  easy  to  see  directly  that  the 
numerators  ?/'(-r),  /;.(•*')  cannot  be  zero  while  (8)  is  not  an  identity. 
One  of  the  two  expressions  ought  then  to  have  a  constant  value 
different  from  zero. 

(17)  It  follows  now  that,  if  ,  .  .2    1    n — ,    ^^  reduces  to  a  constant, 
we  can  make  all  the  transformations 

[cp{x)J  _  [0.(/)]' 

{Ax'  J^Bx-\-Cy      {A  'f  +  B't  -^Cy 

where  A' ,  B' ,  C\  I' ,  m'  have  values  such  that  the  quotient 

t^t) 

I  {A'f^B't-\-C'y 

reduces  to  the  same  constant.  Among  these  transformations  we 
can  find  one  which  is  of  the  first  degree  in  / ;  it  is  sufficient  to  take 

A'^B'^O,     C  =  o,     l'  =  \,     m'  =  f; 


322  LINEAR  DIFFERENTIAL  EQUATIONS. 

then 

A'f(i-ty 


3 


If  we  take  a  proper  value  for  A\  we  shall  be  able  to  make  the  trans- 
formation 


[Ax' -\- Bx  +  CJ      gA'{t-i)' 

which  is  of  the  first  degree  with  respect  to  L     It  is  clear  that  every 

transformation  which  we  can  make  in  the  same  case  will  conduct  to 

the  preceding  followed  by  a  new  transformation,  equally  of  the  first 

degree  with  respect  to  /. 

f  (x) 
If  the  ratio  —  "^  ' —  has  a  constant  value,  we  will  take  in  like 
A'ix—  of 

manner 

ji>j^B'  =  o,    c  =  o,   r  =  h    m'  =  l, 

0W  =  -  -^-^'(^  -  0.    0/ W  =  -  —  ^(3^  -  2). 
Now 


{A'f+B't-\-CJ      4A'{t-i) 

i^A^)  ___      It -2 

{A'f  +  B't  +  Cy      \A'{t-\Y 

The  difference  of  these  is  —-77 ;  if  we  take  a  proper  value  for  A' ,  we 

2,/]. 

can  effect  the  transformation 

[0(-^')]°  _  t 

{Ax"  -\-Bx+  Cy      4^'  (/  -  i) ' 

from  which  we  derive  the  same  conclusion  as  above. 


GOURSAT:  HYPERGEOMETRIC  SERIES.  323 

In  order  then  to  obtain  the  new  transformations,  it  suffices  to 
-combine  the  rational  transformations  which  can  be  effected  for  the 
two  systems  of  values  of  the  constants  A,  B,  C,  /,  m, 

C=o,     A-\-B  =  o,     I  =  h     ^«  =  f  ; 
C=o,     A-\-B  =  o,     /  =  i,     VI  =  I. 

(18)  It  remains  to  examine  the  case  where  there  exists  a  transforma- 
tion of  the  second  degree  with  respect  to  the  two  variables.  Let  x^ 
be  a  value  of  x,  and  /„ ,  /,  the  corresponding  values  of  /.  To  the  value 
/„  correspond  for  x  the  values  x^  and  x^ ;  to  the  value  /,  correspond 
for  X  the  values  x^  and  x^.  If  x^^x^,  the  relation  F{x^,  x^  =  o 
between  two  values  of  x  which  answer  to  the  same  value  of  /  will  be 
of  the  third  degree,  and  we  thus  come  back  to  the  preceding  hy- 
pothesis. If  x^^x^,  the  relation  F{x^,  x^  =  o  is  of  the  second 
degree  and  breaks  up  into  two  relations  of  the  first  degree,  viz., 

_  _  ax,-\-  b 

From  what  we  have  seen  above  this  last  relation  should  have  one  of 
the  forms 

I  X. 


X  \  —  1        x^ ,     X  ^  —       ,     x^ 


■*o  X^  I 

Suppose,  for  example,  .*",  =  i  —  x^.     The  relation  between  x  and  t 

will  now  be 

(2^-  !)»=:/,(/), 

J^  (/)  being  a  rational  function  of  t  of  the  second  degree.     The  two 
differential  equations 


VAx^  +  Bx+C  ^^  ^  VAx^±Bx,±C^^^^ 


x{i-x)  X,{l-X,) 


324  LINEAR  DIFFERENTIAL   EQUATIONS. 

dx Kdx^ 

XI  {i  —  x)""  ~  x/  (i  —  x^)"^ 

should  then  have  for  integral  x  =  i  —  x^;  for  this  we  must  have 
I  =  m,  A  -\- B  =  o;  but  then  we  could  make  the  transformation 

{2X  —  l)^  =  U, 

and   the  proposed  transformation  would  be  a  combination  of  the 
two  following: 

{2x-iy  =  u,     u=/,{t), 
which  is  one  of  the  transformations  indicated  above  (VIII,  X,  XII)., 

(19)  For  brevity  we  will  denote  by  U,  V,  W,  respectively,  any 
one  of  the  transformations  contained  in  the  following  three  tables : 

a (2.  -  ly,  {^)\  (i  +  ^V  (^^  -  0'    (2  -  ty   (I  +  tr 

v.. 


t    I  '  \i  —  tl  '  4t{t  —  i)'  4(1  —  /)'       4/ 
(/»  _  6/  +  I)"    (/'  +  4/  -  4y    (I  +  4^  -  4n' 


w... 


—  16/(1  -tf  —  i6t\i  -  ty     16/(1  -  /)    • 

{9t  -  8)'  (I  -  90'        ^(9-80'    (i  -  /)(!  +  8/)' 

—  27/^(1  —ty  -  27/(1  -  ty  27(1  -  /) '         27/ 

t{t-9r    (/  +  8r(i  -/)  (8/'  -  36/  +  27)- 

-27(1-/)''       -27/'      '    -64/^(1-/)   ' 

(8/'  +  20/  -  I)'    (8  -  36/  +  27/^)'    (27/^-  18/-  I)' 

—  64/(1-/)'  '      64(1-/)      '  64/ 

(/'  +  18/  -  27)^    {f  -  20/  —  8)'    {2f  -  3/'  -  3/  +  2)' 

647        '     64(1-/)'    '      -2'jf{\-ty 

(64/°  -  96/'  +  30/  +  I)'   (/'  +  30/^  -  96/  +  64)' 
—  108/(1  — /)       '         108/*  (I  -  /) 

{t'  -  33^'  -  33^  +  0' 
108/(1  —  ty 

Employing  these  conventions,  all  the  new  transformations  will  be 
found  in  the  following  table: 


I 


GOURSAT:   HYPERGEOMETRIC  SERIES. 

'A+B  =  o,  0  =  0,-] 
LXVI...  \  /=m  =  i,  \.  {2x  -  if  =  V, 

A  =  ±  yu  =  ±  i. 


LXVIL.J 


f         A=:0,    C=0, 

/  =  I,  in  =  ^, 


(I  -  xf 


=  v, 


XXVIII... 


A  =  o,  B  -\-C  =  o, 
/  =  i,  m  =  I, 


{—-)'=  -• 


LXIX. 


(A  -{-B  =  o,  C  =  0,'] 

I 
/  =  ;«  =  !,  ^  (2;r  -  I)'  =  W, 


LXX. 


LXXL. 


LXXII. 


A  =  o,  C  =  o, 

[     X=±r=±±  J 
A  =  o,  B-\-C  =  o,^ 

/=7n  =  §,  [ 

I 

M=±r=  ±i.      J 

A=o,  B  =  o,  1 


1+^ 


2  —  X 


=  W, 


=  w. 


l=m  =  I, 


4x{x-  i)       "^^ 


LXXIII. 


V=±2^X=±^ 

B  =  o,  C  =  o,      "] 

I 
/  =  I,  m  =  f , 

X=  ±1,  ix=  ±v,] 


(2  -  XY 

4(1  -  x) 


=  W, 


325 


326 


LXXIV. 


LINEAR  DIFFERENTIAL  EQUATIONS. 
A  =  C,   B-\-2C  =  O, 

'■  ^^  ^1    ^^^  ^=   3^> 


LXXV. 


.  At  =  ±  I,  A  =  ±  r. 
A  =  C,  B  =  2A, 
/  =  I,  m  =  ^, 


(I  +  ^y 


4x 


=  W, 


LXXVI. 


A   =  —  =    +    V. 
2 


C=  4A,B=  -4A, 
I  =  h  in  =  i, 


±ix=  ±y. 


{A=4C,  B  = 
I  =  m  =: 


aC, 


LXXVII...  \ 


A  =  ±  A<  =  ± 


LXXVIII. 


'^=o,  4^+3^  =  0/ 


/  =  f ,  ;;/  =  f , 


V  =  ±  i,  -  =  ±  yU. 


f    ^  =  o,  ^  =  3C 

2 


LXXIX. 


^  —  T'     ''^    —    3"' 


i 


V  =  ±  i,  A  =  ± 


(;^^  -  6.r  +  i)^ 

—  i6,r(i  —  A')" 
{x"  -  6;r  +  i)^ 

—  i6;i'(l  —  xf 

(x^-\-4x-  4)' 

—  l6;r'(l  —  ;ir) 

(^'  -I-  4;f  -  4)' 

—  i6x\i  —  X) 

(i  +4^-4;r7 
j6x{i  —  ;r) 

(i  +4;t:  —  4xy 
l6;ir(l  —  ;r) 

—  2yx\i  -  x) 

{9-^  -  8)' 

—  27-r'(i  —  x) 

(I  -  9-^y 

—  2'jx{\  —  xy 

(i-9^r 

—  27;jr(l  —  jr)^ 


^7, 

^, 

I/. 
V, 

■-  u, 
--  V, 

u, 
w, 

u, 


GOURSAT:  HYPERGEOMETRIC  SERIES. 


327 


LXXX. 


LXXXI. 


LXXXII. 


'  C  =  o,  aB^-^A  =:  0,  ~ 

/  =  i,  w  =  f , 

(9  -  sxyx 

27{i-x)    -  ^' 
27(1  -  x) 

-  27(1  -  xy  -  ^' 

-  27(1  -  xy  -  ''^ 

(^  +  8y(l   -^)  _^r 

—  27;t-'                        ' 

•  (.•  +  sy'(.  -  -)  _  ^ 

—  274:'                        ' 

x  =  ±h  /^=  ±^- 

'      C  =  o,  B=zA,     - 
l=h  ^«  =  f, 

X 

LXXXIII. 


/  =  -JT-,   ;//  =  2"? 


fA  =  o,  gB-^8C=o, 


(I  +  8xy(i  -  ;^) 

2']X 

(i  +  8^)Xi  -;^) 

2']X 


=  u, 
=  w, 


LXXXIV...  ^ 


t  —  2  >  ^^^  —  T' 


LXXXV...  \ 


r  =  ±  1   -  =  ±  /i. 

^  =  o,  ^  =  8^:', 

/  =  I,  m  =  i, 


r  =  ±  i,  A  =  ± 


/^ 


- 

-  64x\i 

-^) 

-  ^, 

{Sx 

'-  36^ 

-^27r 

=  w, 

— 

-  64x\i 

-^)  ' 

{8x 

'  -\-  20;r 

-  ly 

u, 

— 

64^(1  - 

-^r  ~ 

{Sx 

'-\-  20;r 

-ly 

w, 

— 

64x{i  - 

-^y  ~ 

328 


LXXXVI. 


LINEAR  DIFFERENTIAL   EQUATIONS. 
'  C=0,  9^+8^  =  O, 


t  —   g,    jn    ^^ 


LXXXVII... 


A  =    ±   i     yU=   ±-. 

f    C  =  o,   B  =  SA, 

/  —   2      i,f  —  1. 


(8  -  36^  +  2yxy 
64(1  —  x) 

(8  -  36X  4-  27xJ 
64(1  -  x) 

(8  +  20;r  -  ;ir^)^ 


=  W, 


LXXXVIII...  H 


3 
^=9C  B-\-ioC=o, 

^  =  7)    ^^^  ^=    3^» 


64(1  -  xy 

(8  +  20;t:  -  .r')' 

64(1  -  xy 


=  u, 
=  w, 


\ 


LXXXIX...  ^ 


C=9A,  B-{-ioA=o, 
/  =  !    m  =  ^, 


XC...  ^ 


//  =  ±  4,  -  =  ±  r. 


A  =  C,  B-\-C=o, 


{2'jx'  -  \%x  —  i)' 

64X 
(27;r'  —   i8;r  —   l)' 

64^ 

(;r'+  i8;r  -  27)' 
6zpr' 

(;r^+  i8;r  -  27)' 


=  u, 
=  u, 

=  JV, 


XCI...  ^ 


A.  =  ±  yu  =  ±  r. 

A-{-B  =  o,A=  16C, 

/=7n  =  I, 


A  =  ±/i=  ± 


64;ir' 

(2^»  -  3^'  -  3^  +  2Y 

-  27^X1  -  xy 

{2X'  -  3;i;'  -  3.y  +  2)' 
—  27;ir'(l   —  ;ir)'-' 

{64x'-g6x'-\-^ox-{-iy 

—  io8-i'(i  —  x) 
{64x'-g6x'-{-2,ox-}-iy 

—  ioSx{i  —  -r) 


=  W, 


I 


XCII. 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


(;tr'+  lOx"—  g6x  +  64)' 


=  ±  ^^  =  ±y. 


io8;r'(i  —  x) 

{x^  -j-  30^'^  —  g6x  -j-  i)' 

lO?)X\l  —  x) 


329 

=  u, 
=  w, 


'  A  =  C,  B=i4C,   ~ 
/=l  m  =  i, 

\o^x{i  -  xy 

"(^'-33^^-33^+   !)■' 

A  =  ±  -  =  ±   K. 

I                  4 

XCIII. 


The  transformations  thus  obtained  are  not  all  irreducible.    Thus, 
it  is  clear  that  the  relation 


{gx  -  8)^ 


{9t  -  Sy 


—  27x\l  —  x)  —  2'Jt\\    —   t) 

is  equivalent  to  two  distinct  relations,  of  which  one  is  x  =  f,  and  the 
other  is  of  the  second  order  with  respect  to  each  of  the  variables. 
So,  also,  the  transformation 


{2X  -  If  = 


{2f  -  3/'  -   3^  +  2)' 

-27t\i  -  ty 


breaks  up  into  two  rational  transformations  which  have  already  been 
found  directly : 


2;tr  —  I  =  ± 


2f-  Zf  -3^  +  2 


These  simplifications  are  readily  perceived  in  each  particular  case,  so 
no  more  need  be  said  of  them. 

(20)  Suppose  that  for  proper  values  of  ^,  B,  C,  /,  vt  we  can  effect 
the  transformation  x  =  (pit),  and  thus  pass  from  equation  (2)  to  equa- 
tion (4),  If  for  X  =  0  several  values  of  t  become  equal  to  t^ ,  different 
from   zero   and  unity,  these  values,  as  we   have   seen,  will  be  two, 


330  LINEAR  DIFFERENTIAL  EQUATIONS. 

three,  or  four  in  number.  Suppose  first  that  there  are  two  such 
values,  say  /'  and  t" ,  and  suppose  further  mod.  /j  <  i.  Equation 
(4),  and  consequently  equation  (2),  will  then  admit  the  integral 

t>\^  -  t'f  F{a',  /5',  y',  /')+  t"'\l  -  t'i'F{a',  /?',  y' ,  t"). 

This  function  is  evidently  holomorphic  in  the  region  of  the  point 
xz=o;  for,  if  we  make  the  variable  x  describe  a  loop  about  this 
point,  the  roots  /'  and  t"  will  simply  be  interchanged.  Equation  (2) 
will  then  admit  a  uniform  integral  in  the  region  of  the  origin.  This 
integral  will  be  of  the  form 

(i  _  xy^F{a,  /3,  y,  x)  ; 

and,  denoting  by  ^4  a  suitable  constant,  we  shall  have 

A{i  -  xy'F{a,  fi,  y,  x) 

=  /\i  -  i')F{a',  13',  y',  t')  +  t"'\l  -  t")'  F{a',  f3' ,  y\  t"\ 
Equation  (2)  also  admits  the  integral 

t''\Y  -  t'i F{a\  fi',  y',  t')  -  t"^\i  -  t"fF{a',  /5',  y',  t'\ 

which  is  reproduced,  to  sign  pr^s,  when  x  describes  a  loop  round 
the  origin.     This  integral  will  be  of  the  form 

x^{i  —x)-^^F{a^,  ^,,  y^,  x); 

and  so,  denoting  by  ^  a  constant,  we  shall  have 

Bxi{i  -  xy^^F{a,,  A,  r:.  ^) 

=  /'(i-0'>(«  ,  /3',  /,  n  -  t"\l-t'i' F(,a\  ft\  y',  t"). 

If  the  values  of  /  which  become  =  /,  for  ;f  =  0  are  in  number  greater 
than  2,  it  is  easy  to  form  the  integrals  of  (2)  which,  when  the  vari- 
able describes  a  loop  round  x  =  o,  reproduce  themselves  multiplied 
by  a  constant.  Suppose,  for  example,  that  three  values  of  t  become 
equal    to    /,  when  x  =  o,  and    let   /',   f,  t'"  be   these  three  roots 


GOURSAT:   HYPERGEOMETRIC  SERIES.  33 1 

arranged  in  the  order  in  which  they  present  themselves  when  the 
variable  describes,  in  the  direct  sense,  successive  loops  round  the 
origin.     The  integral 

t'\l  -  t'f'F{a',  ft\  /,  t')  +  t"'\i  -  t'i'F{a\  ^' ,  f,  t") 

J^t"''\l-t"'f  F{a',fi\y\t"'\ 
is  evidently  uniform  in  the  region  of  the  origin  while  the  integral 

t'\^  -  t')''F{a',  13',  /,  n+jY''\i  -  t'i'F{a',  fS' ,  y' ,  t") 

-^jr''\l-f"fF{a',ft',y',t"'y 

reproduces  itself  multiplied  by  j  when  x  describes  a  loop  in  the 
direct  sense  round  the  origin.  We  deduce  now  relations  analogous- 
to  the  preceding.  One  or  several  values  of  /  may  be  zero  at  the 
same  time  as  x.     If   the    value   of  t  which  vanishes  for   -r  =  o  is 

/'  g' 

unique,  the  integral  /  (i  —  0  ^Wy  §\  y' ■,  t)  reproduces  itself,  to 
a  constant  factor  pres,  when  x  describes  a  closed  path  surrounding 
the  origin.     It  ought  then  to  be  equal  to  an  integral  of  the  form 

Cx-^  {i  —  x)-^  F{a,  f3,  y,  x). 

The  case  when  several  values  of  t  are  zero  at  the  same  time  as  x 
requires  a  fuller  explanation.  Suppose  first  that  x  =  0(/),  where  0 
denotes  a  rational  fraction.  Equation  (2)  admits  an  integral  of  the 
form 

and  the  same  is  true  of  equation  (4).  If  we  consider  f  as  the  inde- 
pendent variable,  then  when  /  describes  a  loop  round  i  =  o,  x  wifl 
return  to  its  initial  value  after  having  described  several  loops  round 
x  =  o,  and  the  preceding  integral  will  reproduce  itself  multiplied  by 
a  constant  factor.     We  have  then 

x-^{l  -x)-^F{a,  /?,  y,  x)  =  C't'^\i  -  t')'' F{a\  (3',  y' ,  t'), 

t'  denoting  one  of  the  roots  of  the  equation  x  =  0(/)  which  are  zero 
when   X  =  o.      Since   every   transformation    can    be   led   back   to> 


332  LINEAR  DIFFERENTIAL   EQUATIONS. 

rational  transformations,  the  preceding  conclusion  holds  when  x  is 
no  longer  a  rational  function  of  t. 

The  calculation  of  the  constants  which  enter  into  our  formulae  is 
effected  without  difficulty,  when  for  ^  =  o  we  have  at  the  same  time 

/  =  o:  it  is  sufficient  to  seek  the  limit  of  the  ratio  for  x  =^  o. 

X  -f 

Now  take  the  case  when  for  ;ir  =  o,  /  takes  a  finite  value,  t^,  differ- 
ent from  zero  and  unity.  In  the  formulae  written  above  make 
.;r  =  o,  t  ■=t^\  we  get 

A  =  2tf\i  -  tfF{a',  §',  y',  /,), 

£  =  lim.  ^^"'(i  -tY^W,  /3\  /,  t')-t"'\l-t"YF{a',  (3',  y\  t") 

for  X  =  o,  /  =  /j , 

The  following  examples  show  the  method  to  be  followed  in  each 
•case. 

(21)  Consider  the  differential  equation 

72  f 

Writing  x  =  {2t  —  if,  this  becomes 

/(/  -  i)^  +  [^  +  y5  +  1  -  (2«  +  2^  +  i)/]^  -  4«/?j/  =.  o. 

For  X  =  o  the  two  values  of  /  are  equal  to  ^.     We  shall  then  have 
aF{a,  13,  f,  x)  i 

=  FLa,2ft,a  +  ft-^h^-^^)+F(2a,2^,a  +  ft  +  h^-^^y 
=  F(2a,2ft,a-^rl3  +  h'  +  ^''V/'^2^,  2^  a  +  yg  +  i  i:^- Y 


1 


COURSAT:  HYPERGEOMETRIC   SERIES.  333; 

For  ;r  =  I  one  of  the  values  of  /  is  =  o ;  then 

F{a,  y5,  «  +  /J  +  1    I  -  x)=F(2a,  2A  «+  ^  +i  '  ~  ^. 

This  last  formula  enables  us  to  calculate  a  and  b.     We  have  in  fact 

I  a  =  2F{2a,2l3,  a  +  ^  +  hk\ 

or,  from  the  preceding  formula, 

In  like  manner, 

^  =  ^ip^q:T^(2«  +  1, 2^  +  I,  a  +  /?  + 1, 1). 

In  the  value  of  a  change  a  into  a  -\-  ^  and  /?  into  ^  -\-  i;  we  findl 
then 

^-«r  +  /?  +  ir(ar+i)r(^+i)-       r(^)r(/j) 

(22)  Consider  again  the  differential  equation 


Writincf 


4 
{f  -6t-\-  if 


--  -  -  16/(1  -tY' 
and  at  the  same  time  making 

we  have  the  new  differential  equation 

For  jf  =  o  two  values  of  /  become  equal  to  3  —  2  V2. 


334  LINEAR  DIFFERENTIAL  EQUATIONS. 

Let  /'  and  t"  denote  these  values ;  in  the  region  of  ;ir  =  o  they 
are  developable  in  a  convergent  series  going  according  to  ascending 
powers  of  x^.     Let 

/'  =  3  —  2  ^+  ^  —  I  (3  1^2  —  4);ir*  +  .  .   .  , 
^"  =  3  —  2  VJ  —  V"^  (3  1/2  —  4);iri  _!_.... 

The  integral 

/"(I  -  0^"^(4«,  2^  +  i,  2«  +  I,  /') 

+  /'^"  (I  -  /'0^''^(4^,  2«  +  i,  2a  +  I,  /'O 

is  uniform  in  the  region  of  the  origin.     We  have  then,  A  being  a      J 
constant, 

AF{pc,  ^-  a,i,^)  =  ^'V  -  n'''F{4cy,  2a  +  ^,  2ar  +  |,  /') 

-I-  r\i  -  t"YF{A,a,  2a  +  i,  2«  +  I,  /")• 

In  like  manner,  denoting  by  ^  a  new  constant, 

BxiF{a  J^^,l-  a,i,x)  =  r{i  -  ty'^F{4a,  2a  +  i,  2«  +  |,  /') 

_  /'-(I  _  fJ-'^Fi^a,  2a  +  i,  2^  +  |,  t"). 

Making  ;r  ==  o,  we  get 


^  =  2  (3  —  2  V2Y  (2  V2  -  2y'^F{4a,  2ar  +  ^,  2«  +  I,  3  -  2  V2), 

d 
dt 


B  =  2  V^^i  (3  V2  -  4)  -,  [/«  (I  -  ty^F^^a,  2a  +  i,  2«  +  |,  /)] 


for  /=  3  —  2  Vz- 

Observe  now  that,  for  ;tr  =  00,  one  of  the  values  of  t  becomes  zero. 
The  proposed  differential  equation  admits  the  two  integrals 

/»(l  -  /)-/^(4«',  2a +1,  2«  +  f,  t), 
which  belong  to  the  same  exponent  in  the  region  of  the  points-  =  00. 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


335 


They  ought  then  to  be  identical,  to  a  constant  pres.     We  deduce 
from  them  the  relation 

F{\a,  2a  +  ^,  2ar  +  |,  t) 

which  can  also  be  written 
F{4a,  2a  -{-  I,  2a  -\-  ^,  t) 

=  (I  +  ^)-^^F[a,  a-{-i,2a  +  l  '-^~~~^)- 

In  this  last  formula  make  /  =  3  —  2  V2  ;  then 

F{4a,  2a^^,  2af+f,  3-2  4/2)  =  (4-2  V'2)-^''F{a,  a-}-^,  2ar+|,  l), 

and  so 


A 


'(3  -  2  l/2)(2  -  2  V^2y>       2  4/^r(2^  +  |) 

(4-2|/2y        J  ria  +  i)r{a  +  i) 


I  y     2  V7rr(2a-\-f) 


In  the  same  way  we  have 
/"(I  -  ty''F{4a,  2a  +  ^,   2ar  +  |,  /) 
/(I  -0 


i^ 


r  16/(1 -/)n 

or,  «  +  ^,  2ar  +  |,      .      ,    ^Tr-_ 


L(i  +  O^J 


|/7rr(2ar  +  f) 


i^ 


r(-i)r(2^+f)/--6/+i 


F 


«  +  2,  «  +  f >  l» 


(/^  -6/4-  i)'- 


Taking   the  derivative   of   the  second  member  and    neglecting  all 
terms  which  are  zero  for  /  =:  3  —  2  V2,  we  find 


\B  =  2  V^^ 


-(3  -2  4/2)(2i/2-2y-|" 
(4—2  V2Y  J 

(3i/J-4)(-4  4/i)   r(-  i)r(2^  +  f) 
{4-2Vly       '    r{a)r{a-{-i)  ' 


33^  LINEAR  DIFFERENTIAL   EQUATIONS. 

or,  on  reducing, 

^-^     Hi6J   r{a)r{a  +  i)- 

23.  The  following  formulae  have  been  obtained  in  an  analogous 


way: 

(25)     ^F{a,  /?,  i,  x) 


=F\  2a,  2/?,  a 


-\-ftW-^^)-^F 


I  2a,  2f3,  a 


+  /5+i 


I  —  |/j 


a  = 


2|/^r(a'  +  /?  +  i) 


r(«  +  i)r(ys  +  i) 


i^' 


(26)    ci^(«,  /?,  i,  x) 


=  (i 


—  ;ir)-"/^|  20-, 


I  —  2(3,  a  -\-  I  —  /3, 


v: 


-\-{l-x)-<^F{  2a,  I  -  2ft,  a -^  I  -  /3, 


2v: 

Vx  — 
~~27. 


K-  I         / 
I    —  V  J\ 


C  = 


2  4/^r(a'+l  -/?) 


(27)    ^  i/I/^(a',  /?,  f,  x)  =  F(2a  -  i,2ft-l,a-}-  J3 


F[2a-  1,2J3-  I,  a-\-ft-^, 


2 

I    -    |/,t 


(28)     diGF{a,  /3,  f,  ;tr) 
=  (l  —  x)'"'FI  2a 

—  (l  —  ;f)-»/^|  20-  - 


I,  2  -  2^,  ar+  I  -/?, 
I,  2  —  2/?,  «+  I  -yS, - 


V^' 


;f  —  I  +  -/A 
2  |/^  —  I       / 


2 1/^  — 
1/^—  I  — 


21/, 


;ir 


^  -  i\a  -  i)r(/?  -  i) '    ^  -  r(a  -  i)r(i  -  >s) ' 


GOURSAT:   HYPERGEOMETRIC  SERIES. 


337 


(29)  F{a,  /3,a-{-^  +  i,x)  =  F{2a,2^,a-i-/3+^, 

(30)  F{a,  ^,  a  +  ^  +  h  ^) 


I  —  |/l  — 


I  —  x\ 


I  +  '^^I— -^'\  zr/  /?    I     1  1     /?    1     1      ^I— ■^—  I 


(31)      i^(«',    /?,    «+  /?  +  i  ^) 


=  (  i^i  —  A'+   V—  x)-^'^Fl  2a,  a-\-  ^,  2a -\- 2(3, 


A 


2  V'- 


(32)     F{a,  ^,   a  -^  /3  -  h  x)  

=  (I  -  xyiFf2a  -  I,   2/3  -  I,   a  +  ^  -  i,    '~  ^' ""'Y 


(33)     /^(^,  ^,  a-\-^-h  x)  =  {l  -  x)-i 


F 


^_+^^ 


'F(2a-  I,    a-/3  +  h^  +  /3-h  ^^^=J^ ?Y 

\  y  I  — ;f  +  V 

(34)     F{a,  /3,  a  +  /3  -  i,  x)  =  {i  -  x)-i{  Vi  -x+V^^y  — 
pLa  _  I,  «  +  ^  _  I,  2«r  +  2/?  -  2,   --^ri£^Y 


-\-  ^  —  I,   2a 

(35)     i^(a',   a  +  i   r^   -*■) 

_  /I  +  4/7^ 


(36)     F{a,  «+i    ;.,  ;r)=(l-;^)-/'/2^,  2;.-2«-I,  ;.,  i^i^^^V 

\  21/1— ;ir/ 


(37)   /^(o',  a+h  r,  x)={i-i-  Vxy-'^F (2a,  r-i,  2y-i,  ^^^\ , 

\  i  +  Vx/ 


538 


LINEAR  DIFFERENTIAL   EQUATIONS. 
a    ft    0'  +  /?+  I 


(38)    F\a,ft,'^±^-^,x]=.F 


(39)    FU,  fi 


=  {\    —   2X)F 


2'   2'  2 

«  +  I      yS+  I      0-  +  /:^+  I 


,  4^(1  —  X) 
,A^{\  —  x) 


^  X]  —  {\  —  2,r)-* 

a     a -\-  \     a-\-ft-\-\     4x{x—i) 


F 


'      {2X  -   I)'  J  ' 


«  +  /?+!       J 


,         ^^ 


(40)    i^(^fl',  /?, 

/ ,        / X  r-T         «'  +  /^  1      />  4  ^A^'  —  l) 


(41)     /^(a,  i-^,  r,  x)  =  (i-^)v-'/^ 
=  (i  —  x)y-'  (l  —  2;i-)/^ 


(l/l-;tr+|/-;jr)''J 
y — a    y-\-a — I 


2        '  2 

y  -\-  a    y  -\-  \  —  a 


-,  y,  4x{l—x) 


,  y,  4x{l  —  x) 


(42)     F{a,   I  -  a,  y,  x) 

=  (i  _;^)7-i(i  -  2xY-yF 


y  —  a    y  -\-  I  —  oc  4x{x  —  i)- 


(2^  -  ly 


(43)    F{a,  I  -«,  r'^)=(i  -x)y-^{Vi-x-\-V-xy--—-y 
Fly  +  a-  I,  y  -h  2y-  I,  --=^_}         __ 

L  {vi— x-\-v—  X)  J 


(44)   /^(«,  /?.  2A  ^)  =  (I  -  ^)^  ^ 


'ex     ^       a     ^       I  X 


2  '  4(-^-  I) 


.)] 


=    I 


f)(,-.r°4^/^ 


I  —  a    I  4-x  ,1  ;«; 


2'4(.^-0J' 


(45)     F{a,ft,2/3,x)=[i--       F 


2      '       2 

_2'         2      '  ^^^2'   V2-^-/ 


=(,-.)-»(.  -  f)-'V[,_  f ,  .+i^«,  .+i,  (j^jn, 


GOURSAT:  HYPERGEOMETRIC  SERIES.  339 

(46)     F{a,  A   2/?,  X) 

JL  (I  —  '^i  —  -y) 


=  (i  -  xY'-F 

(47)      /^(^,    /?,    2/?,    ^) 


a,  2^-  a,  ft -^ 


L  (I  -  i^i  -  ^n 

—  4r  I  —  ;t^  -■ 


F 


a,a-l3Jrhft  +  h 


-VT 


I  4-i/i— ^j  J' 


(48)  F{a,   fj,   a-  /3  +  1,  x) 

=  (i  —  xY°-  F    -  ,  ^ - 

2  2 

(49)  F{a,  /3,   a  -  /3  -^  I,  x) 
=  (i+)(l-x)-«-'i^ 


,  a-/3-{-i, 


—  4x 


(I  -  -*f  J 


(50)       F(a.    /J,    «   _  /i+  I,    ;i-) 


.2'    2  "^    ^    ''  (1+^)^]' 


(51)   /^(a',  /3,  a-  ft-\-i,  x)  =  {i  -xy--^{i^xyp—-^ 


(52)   /^(^,  /?,«-/?+  I,  ^) 


(i  +  Vxy^^F 


f,  a  —  ft-\-i,  2a—  2ft-\-i, 


^Vx 

(r  +  4/Iy 


0' 


FormukB  furnished  by  the  Rational  Transformations  of  the  Fourth 

Degree. 

(53)     AF{a,  I-  a,  i,  x) 

=  A{1  -xyF{i-a,  a  +  i,  h  x) 

=  t'\i  -tTFiAoc,  2a  +  i,  2^  +  I,  t') 

+  t"\i  -  t'TF{4a,  2a  +1,2^+  |,  ^-), 


340  LINEAR  DIFFERENTIAL  EQUATIONS. 

(54)     Bx>'F{a  +  1,  I  -  ^,    3^   ;^) 

=  Bx^{\  —  xfFii  —a,  «  +  !,  I,  ^) 

=^  t'\i  _  ^')'V(4^,  2ar  +  i,  2«  +  f,  /') 

_  t"\i  -  t'TFiAa,  2a  +  i,  2«  +  |,  /");. 

/',  /"  denote  those  two  roots  of  the  equation 

■  (/»  _  6/  +  i)^  +  i6t{i  -  ffx  =  o 

which  are  equal  to  3  —  2  V^  ior  x  =  o: 

^'  =  3-21/2+  /^^(3  1/2  -  4);iri  +  .  .  .  , 
/"  =  3  —  2  1/2  —  4/^^(3  V2  —  4)xi  +  .  .  .  , 


(55)  ^i^(«,  i  -  «,  i  x) 

=  A{l  -  x)iF{i  -  a,  a-\-lh  ^) 
=  (-  t'ni  -  trF{4a,  h  2a  +  f,   t') 

+  (-  t")\i  -  t"rF{Aa,  i,  2«  +  |,  Ov 

(56)  BxiF{a  +  1,   3  _  ^^   3^  ;^) 

=  Bx^{\  —  x)^F{i  —  «r,  ar  +  f,  f,  ;«;) 
=  (-  /0"(i  -  ^'r^(4«>  h  2a  +  I,  o 

-  (-  t")\i  -  /'o'^^(4^,  h  2a  +  f,  ^0;^ 

t',  t"  are  those  two  roots  of  the  equation 

(i  +  4/  —  4/7  —  \^t{\  -  t)x  =0 

I  —  V2 
which  are  equal  to for  x  =^  o. 


GOURSAT:   HYPERGEOMETRIC  SERIES.  34 1 

2  4  ' 

l"= ; \-V-  I  —  ;ri+  .  .  .   . 

2  4 

•(57)       ^^(^'    T  —   «^'    i'    ^) 

=  C(i  -  x)iF{i  -  a,  a-}-i,h  x) 

=  t''\l  -  t'TF^^a,  2a  +  i,  4^  +  h  n 

+  t"'\i  -  t"TF{^a,  2a  +  i,  4«  +  h  t"), 

'(58)     Dx^F{a  +  ^,l-  a,  |,  ^-) 

=  Dxi{l  -  x)iF{i  -  a,  a  +  l  ^,  x) 
=  /'-(I  _  ^'yF{4a,  2a  +  1,   4^  +  i    f) 

-  t"'\i  -  t'^Fi^a,  2a  +  i,  4«  +  h  H'. 

■i' ,  t"  are  those  two  roots  of  the  equation 

{f  -^  At-  4)'  +  16/X1  -  t)x  =  0 

-which  are  equal  to  2  4/2  —  2  for  ^  =  o. 

/'  =  2  i^  —  2  +  1/^1(3  4/2  —  4);tr*  -[-..., 
/''  =  2  4/2  —  2  -  4/^^(3  4/2  —  4);ri  +   .   .   .  , 

2  4/^r(2^  + 1)  , —  4V^n2^  +  l) 

^  -  r(a  +  i)r(«r  + 1)'    ^  -  '^        r(«)r(«  +  i)  * 

(59)     ^i^(ar,  a-\-\,\,x) 

=  (I  +  /'r^(4«',   2«r  +  i,   2flr  +  I,  O 


342  LINEAR  DIFFERENTIAL   EQUATIONS. 

(60)     GxiF{a  +  1,  a  +  3,  I,  ^) 

=  (I  +  ty-Fi^cc,  2a  +  i,   2«  +  f ,  t') 

-  (I  +  t'^Fi^a,  2a  +  i,  2a  +  |,  /'Ov 

f,  t"  are  those  two  roots  of  the  equation 

{f  -  6/+  i)'  -(I  +0'^=  o 
which  are  equal  to  3  —  2  1^2  ior  x  =  o. 

/'  =  3  -  2  1/2  +  (3  4/2  -  4)^  +  .  .  .  , 

/"=   3  -  2  1^  -  (3  V2   -   4)^  -f    .    .    .  , 

_       i/^r(2a  -f  f)  _  4  4/^r(2a  +  f) 

^  -  V(a  +  i)r{a  + 1) '    ^  -  r  (a)r(«  +  i) ' 

(61)   £/^(a,  a  +  i,  i,  x) 

=  {i  -  2/'r^(4«',  i  2a +  f,  o 

+  (I  -  2/"r^(4^,  i  2a  +  I,  ^'0, 
(62)     (^;ri/^(a  +  i,  a  +  |,  |,  ^) 

=  (I  -  2/'r^(4^,  i,  2a  +  f,  n 

-{i-2t'y^F{^a,  i  2a  +  I,  t")v 
t',  t"  are  those  two  roots  of  the  equation 

(i  +  4/  -  4/')'  -  {2t  -  i)V  =  o 

I    —  \f2 

which  are  equal  to for  x  =■  o. 


1-4/2         4/2    ^    , 
2  4 


2  '        4  ' 


COURSAT:  HYPERGEOMETRIC  SERIES.  343 

(63)  EF{a,   «  +  i,   i,  X) 

(2  -  ty 

(64)  6^^ii^(«'  +  i  «  +  i  I,  ^) 
■j    i^(4a',  2a  +  i,  4a  +  i  /') 


2  -  /' 


2 

^2   -  /' 


/^(4a,  2a +  i,  4a +  i  O; 


2 

/',  /"  are  those  two  roots  of  the  equation 

{f  +  4/  -  4)'  -  (2  -  /)';i^  =  o 
which  are  equal  to  2  \^  —  2  for  ,r  =  o. 

/'  =  2  V2  —  2  +  (3  i/2  —  4)x^  +  .  .  .  , 
t"=  2V2  —  2  —  {2,V2  -  4)x^  +  .  .  .  . 

(65)  HF{a,  i  -  a,  f,  x) 

=  H{i  -  xyF{l  -  a,  a-\-hh  x) 
=  t'\^  -  t'YF^a,  h  2a  +  f,  /') 

+  r<^(i  -fr^{4a,  h  2a +  i,  t") 
_|_  t"'\i  -  t"yF{4a,  h  2a  +  f,  t"') 

+  /iv<^(l  _  /iv)-^(4^^  1^  2«  +  I,  /-), 

(66)  Kx^F{a  ^^,:^-a,^,x) 

=  Kx^{\  -  xJFii  -«',«  +  f,  f,  x) 
=  r{i  -  tyF{4a,  h  2a  +  f ,  /') 


_  ^/-  I  t"\i  -  ffFiAa,  h  2«  +  I,  n 
t"'\i  -  fyF{4a,  i,  2a  +  f ,  /'") 


+  |/iri  /v«(j  _  ryF{4a,  h  2a  +  |,  r); 


344  LINEAR  DIFFERENTIAL  EQUATIONS. 

t\  t",  f",  f^  are  the  four  roots  of  the  equation 

{2t  —  ly  +  i6t  {i  —  t)  X  :=  o, 

taken  in  the  order  in  which  they  present  themselves  when  the  vari- 
able X  describes  successive  loops  round  the  origin  in  the  positive 
sense. 

''  =  *  +  7j(~'4 +*'^  ''"B-^  +  •  •  •  ' 


A  =  (A)  a    cos  -  -f-  V—  I  sin  -         „,    .,./ 1    IS     • 

(67)  HF{a,  a  +  i,i,^)=  (i±itj£!)"  p^^^,  i,  2a  +  |,  f) 

+  (I  +  4/-  -  4/-^)-i^(4«,  i,  2«  +  f,  t% 

(68)  Z;riF(«  +  i,  a+f ,  f ,  ;r)  =  (l±4^Z:4^)^''  ^(4^^  1^  2^  _|_  3 ,  f) 

(i  -X-At'"  —  At"'-'y'- 
-  \       \^  ]    ^(4«.  i  2a  +  I,  /-) 


GOURSAT :  HYPERGEOMETRIC  SERIES.  345 

/,  t" ,  t"'y  f''  are  the  four  roots  of  the  equation 

i2t  -  ly  -  x{i-^  4t  -  4/7  =  o. 


Thus 


V2 


V-  I 

\2 


V2 


V-  I 

'     V2     ^ " '' 

(69)  F{a,  a  +  h2a-{-lx)={l-  x)iF{a  +  ^,  «  +  3^  2or  +  f ,  x) 

(r  +  At'  -  4\^-    , 
=  i— ^^^^ — -j  ^(4«,  2a +  i,  4.,  +  i  o; 

t'  being  one  of  those  two  roots  of  the  equation 

i6f  (i  -  t)-^x  {f  +  4^  _  4)«  =  O 
which  are  zero  for  ;r  =  o. 

(70)  F{a,a-\-\,2a-\-i,x)  =  {i-xYF{a  +  ^,a  +  l   2a -{- ^  x) 

/'  is  one  of  those  two  roots  of  the  equation 

16/'  (i  —  t)  —  x{2  —  ty  =  o 

which  are  zero  for  x  =  o. 


346 


LINEAR  DIFFERENTIAL   EQUATIONS. 


(71)  F{a,  a  +  i,  2a-\-i,x)^{l  —x)kF{a-\-i,  a  +  i,  20- +  |,  ;ir) 

=  {f  -6t-\-  I)-  F{4a,  2a  +  ^,  2a  -f  I,  /)  ;. 

t  denotes  that  root  of  the  equation 

16/(1  —  f)  +  x{f  -6t  ^  if  =  0 
which  is  zero  for  x  ^=  o. 

(72)  F{a,  a  +  h2a^^,x)  =  {l-  x)iF{a  +  ^,  o^  +  |,  2«  +  |,  x) 

=  (I  +  4/  -  ^fy^Fi^a,  h  2a  +  f,  /) ;. 

/  denotes  that  root  of  the  equation 

16/(1  —  /)  —  (i  4-  4/  —  4tyx  =  o 
which  is  zero  for  ;ir  =  o. 

(73)  F{a,  or  +  i,  2^  +  I,  ;f)  =  (I  -  xyF{a  +  i,  «  +  |,  2a'  +  f,  x) 

=  (I  +  />»/"(4«,  2«  +  ^,  20-  +  I,  /) ; 

/  denotes  that  root  of  the  equation 

16/(1  -  ty  -  x{i  ^  ty  =  o 

which  is  zero  for  x  =  o. 

(74)  F{a,  «r  +  ^,  20-  +  !,  x)  =  {l  —  xyF{a-\-^,  «+f,  2a'4-|,  x) 

=  (i   -  2^y-F{4a,i,2a-\-l^); 

t  denotes  that  root  of  the  equation 

16/(1  —  f)  -^  x{2t  —  \y  =  o 

which  is  zero  for  x  =  o. 

Fonmilce  furnished  by  the  Inverse  Transformations. 


(75)     F{4a,  2a  +  I,  2a^lx)  =  {i-xy-^<^F{l-2a,i,  2a -\- I  x} 


(;ir'  —  6x-\-  ly^'^F 
(l  4-;ir)-4-/^ 


,  „  \6x(i  —  xy 


GOURSAT :   HYPERGEOMETRIC  SERIES. 


347- 


{:j6)     /^(4«,  2a  +  i,  4«  +  i,  ^)  =  ( I  -  ^)'  -  '"FiT-a  +  \,  h  Ace  + 1,  ;ir)< 


4  —  4x  —  X' 
4 


i^ 


O',  a  +  i",  2a  +  |, 


—   \6x'{\   —  ;ir)" 


or,  a  +  ^,  2a  +  I, 


l6;ir'(l  —  ;ir)- 


{77)     F{4a,^,  2a  +  |,  ^)  =  (l  _;r)i--i^(2a  +  ^,  |-2a,  2a  +  f,  ;r) 
(I  +4^-  4^r-i^[a,  a  +  i   2a  +  |,  ^^^J  ,. 

(I  -  2^)- -i^ [a,  a  +  i,  2a  +  I,  ^^-^'2^]  ' 


Fonmilcs  furnished  by  the  Rational  Transformations  of  the 
Third  Degree. 

(78)  A,F{a,  I  -  a,  i  ^)  =  Ail  -  ^)Ji^(i  -  a,  a  +  1    1    x) 

.       +  t"-{,  _  /'0V(3a,  3«  +  i  4^  +  I,  /'Ov 

(79)  5,^/^(a  +  h  !-«',  f,  -^)  =  ^.(i  -  x)lF{i  -a,a  +  l^,  x) 
=  ^''\l  -  t'YF{za,  Sa  +  1    4a  +  I,  /') 

-  ^"'v  -  nm^,  zoc  +  i, 4«  +  i  t")- 

t' ,  t"  denote  those  two  roots  of  the  equation 

(9/-  8y+  27t\\  -  t)x  =  0 
which  are  equal  to  -f  for  ;ir  =  o : 


8  4/J 


/"=!-  i/^~F^;^_j_  .  .  .^ 


■      ^^'^    ra  +  *)r(a  +  AV   ^'-^     '^^^""rfa^rra-i-ii 


r(a)r(a  +  i) 


348  LINEAR  DIFFERENTIAL  EQUATIONS. 

(80)  A,F{a,  \-  a,h  x)=  Ail  -  x)lF{i  _  «,  or  +  1    i,  x) 
=  /'a(i  _  ^y<^F{sa,  3a  +  i  2«  +  I  t') 

+  t"\l  -  t'T^ila,   la  +  i,   2«  +  I  t"), 

(81)  B,x^F{a  +  ^,  I  -  «,  I,  ;,)  =  B,x\i  -  x)iF{i-a,  a+|,  f,  ;»;) 
=  /'Xl  -  t'TFila,  la  +  i,  2^  +  i  t') 

-  t"\^  -  t'T^iza,  la  +  i  2a  +  I,  ^-); 
./',  t"  are  those  two  roots  of  the  equation 

{i-^tY-^27t{i-tYx^o 

which  are  equal  to  ^  for  ;r  =  o : 

8  V3 


8  1/3 


(82)  A,F{a,  \  -  a,  h  ^)  =  ^.(i  -  ^W{^  -  «,  «  +  i,  i  ■^) 
=  (-  t'YFila,  i  -  a,  2«  +  I,  O 

+  (-0"^(3^.  i-a,  2a  +  i  O; 

(83)  ^,;ir*i^(a  +  i  |-  «',  f,  ^)  =  ^.^^l- ^)*^(l-  «-  «+i  f.  ^) 
=  (-  t'fF{la,  \-a,   2«+  i  O 

-{-t"rF{la,  \-a,  2a+i    O; 

/,  /"  are  those  two  roots  of  the  equation 

(i  +  8/y(i  —0-  ^7^^  =  O 

which  are  equal  to  —  ^  for  ;ir  =  o: 

t'  =-^-V-i-^x^  +  ..  ,, 


t"=  -i  +  V-  i-^;iri  + 


GOURSAT:    HYPERGEOMETRIC  SERIES.  349» 

(84)  F{a,  \-a,\.x)^{\-  xfF{\  -  a,  a  ^  \A,  x) 

=  {i-tyF{3a,  i-a,  i /),. 

(85)  F{a  +  h  f-^,  f>  x)  =  {i-xyF{i-a,  ^  +  f ,  |,  x) 

=  {i-^y-^'-^^F{sa  +  h    56-a,  f,  /);. 
9  —  01 

t  being  that  root  of  the  equation 

(9  —  8/)v  —  2^{\  —  t)x  —  o 

which  is  zero  for  ;ir  =  o. 

(86)  F{a,  \-  a,\,  x)^{\-  x)\F{k  -a,  a^\,\,  x) 

87)     -^(«+i  f-  ^,  I,  x)  =  {l-x)^F{l  -a,    a  +  f,  f,  ^) 

9       ^ 
/  being  that  root  of  the  equation 

{t-gft  ^  27(1  -  tyx  =  0 

which  is  zero  for  x  =  o. 

(88)  C,F{a,  a+hh^)=  (^— )'V(3^,  3a  +  i  4«  +  f,  O 

+  i^^-^j    -^(3«,  3«  +  i  4«  +  i  ^'0. 

(89)  B.xiFia-^h  a+|,  |,  ^)=  (^^^  V(3a,  3a  +  i  4«  +  I,  O 

-  (^-^)"^^(3«,  3«  +  i  4«  +  I,  ^") ;: 
f,  t"  are  those  two  roots  of  the  equation 

(9^  -  8)^  +  {V  -  4)'^  =  o 


.350  LINEAR  DIFFERENTIAL   EQUATIONS. 

which  are  equal  to  f  for  ;ir  =  o. 

.(90)     C,F{a,  a^\,h  X)  =  (I  +  V'YFila,  3«  +  i,  2«  +  f,  /') 

+  (I  +  It'y^F^la,  3«  +  i  2^  +  I,  /"), 

(91)    Z),^/^(«  +  i  «+|,  f,  ^)  =  (i  +  30^"^''(3^,  3«  +  i  2«+iO 

-  (I  +30^"^(3«^,  3«  +  i  2^  +  1,  O; 

/,  /"  are  those  two  roots  of  the  equation 

which  are  equal  to  \  for  ,r  =  o  : 

*  —  -?  n^    81        ^  '  '  *  ' 


i 


(92)  C,F{a,  «  +  i  i  ;r)  =  (I  -  A^^F^la,  |  -  «,  2«  +  f ,  /') 

(93)  A^  /^(«  +  i  «  +  i  I,  -^)  -  (I  -  At'Y  F{^a,  ^-a,  2^+1,  /') 

-  (I  -  4t"yF{za,  i  _  «,  2a  +  I,  /-) , 

/',  t"  are  those  two  roots  of  the  equation 

(i  +  zff  (I  -  /)  -  (i  -  4tyx  =  o 


GOURSAT:  HYPERGEOMETRIC  SERIES.  251 

.which  are  equal  to  (—  J)  for  ^  =  o: 

^'  =  -i-^^*+  .  .  .  , 

<94)        F{a,  a  +  i,  i  -x)  =  [i-  ~f  F{3a,  i  -  a,  i,  f). 

(95)  F{a+h  a+l  |,  ^):=(i-|)'"'^'' ^/r(3^_|_i  f  _  ^^  3,  ^) . 

/  denoting  that  root  of  the  equation 

(9-  8/)V  +  (3  -4tyx  =  o 
which  is  zero  for  ;r  =  o. 

(96)  F{a,  a-}-^,hx)  =  [l+  -]     Fisa,  «  +  i,  J,  t), 

/  being  that  root  of  the  equation 

{t  -  9)V  -  (^  +  3)^^  =  o 
which  is  zero  for  x  =:  o. 

(98)  E,F{a,  l-a,lx)=  F,{i  -  x)iF{i  -  «,  «  +  1  |,  x) 

+  (i  -0"^(3^,i-^,iO- 

(99)  C;,;r^  F(«  4-  ^,  1  _  ^,  4,  ^)  =  ^^  (j  _  ^)i^^  7^(  I  -  ^,  «.+ 5 ^  I,  ;r) 

!/',  /",  f"  are  the  three  roots  of  the  equation 

(3  -40' -27(1  -  0^  =  o, 


352  LINEAR  DIFFERENTIAL  EQUATIONS. 

taken  in  the  order  in  which  they  present  themselves  when  the  vari- 
able X  describes  successive  loops  round  the  origin  in  the  direct  sense: 

p  3  ^^rji)  _  -9i/^r(|) 

(lOO)     H,F{a,  i-  a,l  x)  =  //.  (i  -  xf  /^(f  -  a,  a -\- H  x) 

+  t'-'Fi^a,  ^-a,2a-^l  t") 

(lOi)     K,x\F{a^\,  i-a,  f,  a')  =  K,xi{i-xy F{i-a,  «  + f,  A  ;,;) 

=.  rFisa,  I-  a,  2a -\- I  t') 
+y7'-/^(3^,i-a,  2«  +  f,0 

/',  ^"j  /'"  are  the  three  roots  of  the  equation 
(i  —  4/)'+  2'jtx  —  o: 

X  V2 

t"'  =  i  +  ^-^-rxi-^  . . . , 

jr     (.y  3r{^)r{2a  +  i)  ^    nr(|)r(2^  +  f) 

^'  -  ^^^  r(«  +  i)r{a  + 1) '   ""'  -     ^^^  r(a)r(«  +  i)  ' 


GOURSAT:   HYPERGEOMETRIC  SERIES.  353 

(102)  H,F{a,  ^-a,\,x)  =  Hil  -  xfF(^,  _  ^,  «  +  i  |,  ^) 

=  {-tyil-tJ'^Fij.a,   3«+l    2«  +  |,   t') 

(103)  K,xiF{a  +  i    i  -  or,   f,   ;ir) 

=  ^,^*(i  -  ;r)i/^(i  -  «,  «  +  f ,  i,  x) 

=  (-  /'^i  -  ^'^-^(3«',  3«  +  i  2a  +  I  i') 

+  j\-  t"Y{^  -  t'YI'ilcy,  Za  +  i  2«r  +  f,  t") 

+y(-  t"'Y{i  -ry-^Fisa,  3a  +  i  2a  +  f,  O;- 

^',  /",  /'"  are  the  three  roots  of  the  equation 

(3/+iy-  27/(1  -tyx  =  o: 

2V2 

/"  1        2^2. 

s  / — 
2  i  "^ 

(104)  L^F{a,  «  +  1    I,  x)  =  (g-  ^ty-t'-F{za,  ^  -  a,  ^,  t') 

+  (9  -  ^t'y'^t"'^F{ia,  i-a,  h  t") 
+  (9  -  ^t"yH"''^F{ia,  \  -  ex,  i,  t"y 

(105)  M,xhF{a  +  I,  or  +  f,  f,  ^)=  (9_8/')-/'ai7(3^^  |_^^  1    ^/^ 

+yX9  -  8/)-/-«/^(3a,  i  -  a,  1    /-) 

+y(9  -  ^t"yH""^F{7^a,  \-oc,h  n> 

t',  t",  t'"  are  the  three  roots  of  the  equation 

(4/-  3)'-  (9  _  %tytx  =  o: 


354  LINEAR   DIFFERENTIAL   EQUATIONS. 


3  V2 


?  +  -TT^  jX^-\-  '   •    •  , 


8 


3  / —  « 

(106)  Pi^(a,  a  +  i,  I,  X) 

=  (I  +  80-^"(i  -  /')"^(3«,  i  -  «',  2«  +  f ,  /') 
+  (I  +  8/'0-(i  -  t"YF{za,  ^-a,2a^l  t") 
+  (I  +  8/"0^'^(i  -  t"'YF{la,  ^-  a,  2a  +  l  t"'\ 

(107)  Qx^F{a  +  1,  «  +  I,  4,  ^) 

=  (I  +  8/')-(l  -  tyF{la,  J-  -  «,  2«  +  f,  /') 
-\-j\l  +  8/"Mi  -  t"YF{la,  ^-  a,2a  +  |,  /") 

+y(i  +  8/-)^"(i  -  ^"0-^(3^,  1  _  «,  2«  +  i  /-); 

/',  /",  /'"  are  the  three  roots  of  the  equation 

(i  -  ^ty  -  (I  +  ^t)\i-t)x  =  o: 

■^  V'2 

^-  r(«  +  i)r(«  +  ly  ^  rHr(«+i)  ' 

(108)  PF{a,  «r  +  i  f,  ^)  =  (I  -  9/')^''^(3«',   3«  +i.  2.*'  +  |,  /') 

4.  (I  -  c^t"YF{ia,    3«  +  i    2«  +  f,  /") 
+  (I  -  9^"0^"^(3«.   3^  +  ^>   2a  +  I,   r"). 


GOURSAT:  HYPERGEOMETRIC  SERIES.  355 

<I09)      (2^ii^(«  +  i,  ar  +  f,    i    x) 

=  (I   -  c^t'YF{la,    3«  +  i    2a  +  f,    /') 
+  j\l  -  9/'0^"^(3^,    3«  +  i    2a  +  f,    ^") 

y,  t" ,  t'"  are  the  three  roots  of  the  equation 

(3^+iy-(i  -90^^  =  o: 

^'   =  -  i  +  ^-  A-i  +  .  .  .  , 

/-  =  -  i  +  — -y^-4  + .  .  . , 
2V2 

(no)    F{a,  a+i  2a+|,  ^)  =  (i  -^)^/^(a  +  i   a  +  f,  2a  +  |,  ^) 

=  (^  -  t)'"^^3«'  3''  +  *'4^  +  *' ^'^' 
/'  is  one  of  those  two  roots  of  the  equation 

27^(1  -  /)  +  (9/  -  ^)^x  =  o 
which  are  zero  for  x  =  o. 
(Ill)     F{a,a  +  h  2a-^lx)  =  {l-x)iF{a+^,a-^l  2a  +  |,  ^-) 

=  (i  +9"(i  -  0'^^(3«.  a  +  i,  4^  +  1,  O; 
/'  is  one  of  those  two  roots  of  the  equation 

27^"  +(^  +  8)'(i  -  ^)x  =  o 
which  are  zero  for  x  =  o. 

<II2)    F{a,a  +  h  2«r  +  f,  x)  =  {l  -  x)h F{a -}- ^,  a  +  f ,  2a +  f,  x) 

=  (i  -  9/)-i^(3a,  3a  +  i,  2a  +  f, /); 


356  LINEAR  DIFFERENTIAL   EQUATIONS, 

t  being  that  root  of  the  equation 

27^(1  —  tj  +  (i  —  ^ffx  —  o 
which  is  zero  for  ;ir  =  o. 

(113)  i^(flr,  «  +  i,2^  +  f,  ;r)  =  (l  -  xfF{a^\,  «  +  f ,  2a +  f,^)' 

=  (I  +  8/)- (I  -  tYF{:^a,  i  -  a,  2«  +  f,  /) ;, 

t  being  that  root  of  the  equation 

27/  —  (i  +  8/)'(i  —  t)x  =  o 
which  is  zero  for  ;ir  =  o. 

(1 14)  F{a,  «  +  i   2a  +  f ,  ^)  =  (I  -  xfF{a  +  i  a  +  f ,  2a  +  f ,  x)^ 

=  (l  -  ^)  'V(3a,  la  +  i,  4«  +  i  t')  ;;        ^ 

/'  is  one  of  those  two  roots  of  the  equation 

27^(1  -  /)-(4  -  Zt)\x  =  o 
which  are  zero  for  ;jr  =  O. 

(115)  F{a,  a-^\,2a^\,x)^{\-  xyF{a^\,  a  +  f ,  2a  +  %,  x)^ 

t'  is  one  of  those  two  roots  of  the  equation 

27/'  —  {\  —  tJx  —  o 
which  are  zero  for  x  ^^  o. 

(116)  i^(a,  a  +  i   2a  +  |,;^)  =  (i  -^)^i^(a  +  ia+i  2a+|,^>      f 

=  (I  +  3^)^"  ^(3«,  3«  +  i,  2a  +  f ,  /) ;. 
/  being  that  root  of  the  equation 

27/(1  -/)'-(3/-j-i)'^=Q. 


I 


G  OUR  SAT:   HYPERGEOMETRIC  SERIES. 
■which  is  zero  for  x  ^=  o. 


357 


<II7)    F{a,  a  +  i   2«  +  f,  ;r):=(i-^)i/^(«  +  iar  +  |,  2a -^^,x) 

=  (I  -  ^typila,  I  -  a,  2flr  +  f ,  0  ; 

^  being  that  root  of  the  equation 

27/  +  (i  —  4tyx  =  o 
which  is  zero  for  ;r  =  o. 


Formulcs  furnished  by  the  Inverse  Transformations. 

<ii8)     F{ia,  2>oc  +  h  A(^ -^  h  ^) 

—  27;r'(i  —  ;r)" 


2a      1— 


8  / 


or,  «  4-  i,  2a  +  |, 
fl',  a  -j-  ^,  2ar  +  |, 


(9^  -  8)^       J  ' 
27;ir'(l  —  xy\ 

(3^-4)^     J' 


(l  19)     i^(3a,  3  a+  i  2a  +  I,  ^) 

=  (i  —  ;r)3-4a/r(i  _  ^,  5  _  ^^  2ar +f,  ^) 

(I  _  gxy^'^F^a,  «  +  i  2a  +  f,  ~  ^"i  _  9;^' 
(i_|_3^)-3a/r    ac,   a+  h  2a  +  f ,  ^^^^  ~  ^"^       ; 

(120)     /^(s^,  «  +  1  4^  +  I,  ^) 

=  (I  -  ;tr>/^(«+  i  3«  +  i  4«+  I,  ^) 

('-4)    ^L^' «  +  *' 2«+i  (j:r^J' 


358  LINEAR  DIFFERENTIAL  EQUATIONS. 

(I2l)     F{la,  \-  a,   2«  +  |,  x) 

=  (I  -  xyF{\  -a,   3ar  +  i    2a  +  f ,  ;^) 


(i  +8;r)— (i  —xY'^F 


«.  «+2>  2a +  f, 


27;ir 


(i  +  8,r)Xl-^). 


J 


(i  —  4x)-3''F 


',  a-\-  I,  2a -\-  f , 


27;ir 


(I22)      /'(Sfl',   ^  -  ^,   1    ;^) 


(123)     /^(3a',  a  +  i,  i  ;r) 

=  (l  _^)J-4a/7(^_  30-,   ^-  «',   1    ;^) 


(I 

-^Y 

2a^ 

(■ 

+r 

-3a 

F 

1         1     (^  -  9)°^ 

I     1      1     (^  -  9)'^" 
«.    «-t-^,    2,     ^^  +   3)' J    ' 


(124)       i^(3«'+l     !-«',!,    4,') 

=  (l  -  ;r)^— i^(i  -  3«r,   a-  +  |,  f,  ;ir) 


I-f)(l--)— ^^ 


9^  V  3 


'a^  +  2>  I  —  «^,  |, 


(9  -  ^xYx- 
27(1  --v)J 
(9  -  8,r)';t-~ 
(4^  -  3f - 


(125)     F{ia+h  «'+i  I,  ^) 

=  (i  —  xf-^''F{i  —  sa,  i  —  a,  f,  x) 


I  --j(i  -^)— '/^ 


^  +  2>    I  —    «^,    f , 


{x  -  gfx 


27{i-xyA 
{x  —.gfx 


■-in'+ir'H"+*- "+*•*•  (-+3) 


GOURSAT :  HYPERGEOMETRIC  SERIES. 


359 


Formul(B  furnished  by  the  Inverse  Transformations  of  the  Other 
Rational  Transformations. 


(126)     F{^a,  4«+i»  6a+^,  x) 

=  (i  —  x)^-^'^F{2a-{-\,  2a-\-i,  6a-{-i,  x) 


1-7  —  36-^'+  8^' 


27 


F 


a,  a-[-^,  2af+|, 


-64x\i  —  x) 


I  -— )      F 
9 


oc,  oc-\-  i    2ar+f, 


(8;ir^-36;r4-27/J' 
64;r'(i  —  xy 

i^-^^xYv 


(127)     i^(4ar,  4ar  +  ^    2ar  +  f,  ^) 

=  (l  —  xf-^''F{^  —  2a,  I  —  2a,   2a  -\-  |,  x) 


{i  —  20x  —  Sx')-^''F 
(i  +  Sxy^'^F 


[a    a^^    -^4-5        -6441-^y 


,  «r  +  1,  2«  +  ^,  ~(i  +  sxy  _ 


(128)     ^(4«',   2«r+ I-,  6a +^,  x) 

=  (l  -  ;ir)iir(4ar+i    2«  +  i,  6«  +  i,  ;i;) 
,■27  —  iSx  —  X 


27 


i^ 


i--j      (i-^)-«/^ 


64;?' 


l8;ir  —  27) 
64^'" 


(;.-9)Xi-^)J' 


(129)     ^(4«',  i  —  2a,  2a  +  f,  x) 

=  (i  -  ^)^F(4a  +  i,  f  -  2^,  2«  +  f,  ^) 


64jr 


(I  +  i8.r-  27x')-^-F^a,  a -\- i,  2a-\-l  JJ^^l8;,- 27XJJ  ' 

64X 


(i  —  9;ir)-3-(i  —  x)-''F    a,  a-{-^,  2a +  |-, 


(130)     F{4a,  i  —  2a,  |,  ;r) 


(9^-ir(l-^)J' 


r  o   (8  —  g^'T^i 


—  ^6x  4-  27;tr' 


/^ 


«'»     ^-\-    2>    l» 


—(8  —  9.y)^;tr 
(8  -  36X  +  27;^=)= 


-]^ 


360  LINEAR  DIFFERENTIAL  EQUATIONS. 

(131)     F{/^a,2a^\,l,x) 


(l  -;tr)-3a/7 

—  20X  —  X 


^»  ^  ^i  T* 


—  {x  +  8)V 
64( 


(x  +  8)';r 


(132)     F{4a -\- ^,  ^  -  2a,  i,  x) 


(i-^)-'^-^(i-^)^ 


«^  +  i,  i  —  «» I, 


8  —  36;ir+ 27Jir'\-^«-3  l'  9;ir\ 


F 


(x  +  h  oi-{-hi' 


(8  —  9.r)°;tr 
64(1-^)  J' 

-  (8  —  9;ir)'';tr 

(8_36;^+27;.r7 


]- 


(133)     F{4a-\-i,2a-]-H,x) 


(I    _;^)-3a-l(l_|.|)/7 

I 8- )  V^  +  Sr     ^+i>^+if>v_2o;r-8yJ' 


(134)     F{6a,  2a  +  1,  4a  +  f,  x) 
=  (l  -  ;i;)^-4-i^(2a'  +  i,  |  —  20^,  4«  +  |,  ;ir) 
/2  — 3;tr— 3;tr''4-2-^' 


F 


a,  a-\rh  2a+f» 


—  27;ir'(l  —  xj 

{2x'-3x'-^X-\-2) 
2'jx''  {\  —  ;ir) 


■] 


r  ,      27X  {\  —  xy  ~\ 

{i-x  +  x'^)-^'^F[a,a^h^<^-\-h  ^(^-^l^j^iyy 


(135)     /^(6ar,  I  -  2ar,  2a  +  I,  ^) 

=  (I  -  ;ir)^--«i^(4a  +  i,  f  -  4«,  2a  +  |,  ;«r) 


I  +  30-r  —  96;^*  -f-  64Jtr')- 


/^ 


f,  «  +  i»  2«  +  f, 


—  I08;tr(l  —  x) 


IVJ' 


(64;r'  —  g6x'  +  30;ir  +  i)* 

io8;ir(i  —  ;tr) 
(l  -  16^  +  l6;ir^)-  3^/r^^,  a,  _^  ^,  2«r  +  I,  ^^  _  ^5^  ^  ^^^^yj  : 


GOURSAT:  HYPERGEOMETRIC  SERIES. 


361 


'(136)     F{6a,  4a  +  1  2«^  +  f ,  x) 
=  {l-  ;r)i-8-/^(5  _  40',  I  -  2a,  2a  +  f,  ;ir) 


(i  +  i4x-{-x')-^''F 


a,  a-\-i,  2a+f , 


108^(1  —  xy 


—  lo8;f  (I  —  xy 
a,  a -{- i,  2a -\-  ^,   (^.  _^  14^  +  1)^  = 


(137)     7^(6^',  4«  +  i,  Bar  +  1  ^) 
'  /64  —  g6x  -\-  30^''  +  x' 


64 


i^ 


a,  a  +  i,  2a -{-^, 


io8;r*(i  —  x) 


]• 


(64  —  g6x  +  30^'  +  x"") 
/i6-i6^  +  .ry3a     r  -io8^-(i-^)n 


The  transformations  which  we  can  effect  when  two  of  the  three 
elements  a,  /?,  y  are  arbitrary  have  been  completely  given  by 
Kummer.  In  the  case  where  a  single  element  is  arbitrary,  he  has 
indicated  some  particular  cases  of  the  rational  transformation  of  the 
fourth  degree  and  a  certain  number  of  irrational  transformations. 
The  other  rational  transformations  and  the  greater  part  of  the  irra- 
tional transformations  above  seem  to  be  new. 


CHAPTER  VIII. 

IRREDUCIBLE   LINEAR  DIFFERENTIAL   EQUATIONS. 

Some  properties  of  these  equations  have  already  been  noted  in- 
Chapter  IV,  but  we  shall  study  the  question  in  a  rather  more  gen- 
eral manner.  Before  entering  into  the  study  of  these  equations 
from  the  modern  point  of  view  which  requires  a  knowledge  of  the 
group  of  substitutions  belonging  to  a  given  linear  differential  equa- 
tion, we  will  give  some  general  theorems  concerning  irreducible 
differential  equations  taken  from  the  memoir  by  Frobenius*  pre- 
viously referred  to. 

In  all  that  follows  we  will  assume  in  <  n.  Denote  by  P  the 
operator 


dx""        "^   dx 
and  by  Q  the  operator 


d'"     .         d 


■m-i 


+  ^>  :7-;;7-;  +  •  •  •  +  ^» 


dx'"     '    ^    dx 

where/  and  g  are  uniform  functions  of  x. 
The  differential  equation 


dy   ,      d"-y  , 


*  Frobenius  :  Ueier  den  Bcgriff  der  Irreductibilitdt  in  der  Theorie  der  linearen 
Differentialgleichungen.  Crelle,  vol.  76,  p.  256.  Frobenius  refers  to  the  memoir  by 
Libri  in  Crelle,  vol.  10,  p.  193,  and  also  to  the  "  Note"  by  Brassinne  in  the  Appen- 
dix to  Vol.  II  of  Sturm's  Cours  d' Analyse. 

362 


IRREDUCIBLE  EQUATIONS.  363 

is  reducible  or  irreducible  according  as  it  has  or  has  not  integrals  in 
common  with  an  equation  of  lower  order,  say 


Let  n  —  m=^  I,  and  for  brevity  write  Q  instead  of  Qy;  no  inconven- 
ience can  arise  from  this  abbreviation,  as  it  will  always  be  clear 
whether  we  mean  by  Q  the  operator  or  the  differential  quantic 
which  is  the  left-hand  member  of  (2).     Form  now  the  derivatives 

dQ.      d^  d^ 

dx '      dx^  '  '      dx^ ' 

and  from  the  equations  so  obtained  find  the  values  of 

d"y     d"'+y  dy 

dx^'     dx"'+''     '  '  '  '      dx" 
and  substitute  these  in  (i);  we  have  then  an  expression  of  the  form 

d'O  d^-'O  d'-^O 

where  r„,  r^,  r^,  .  .  .  ,  r^  are  uniform  functions  of  x,  and  R  is  a.  linear 
function  of 

d^y       d^'^y  dy  _ 

-dx^'     d^'     •  •  "     -di'     ^         (A<^;^-i), 

having  for  coefficients  uniform  functions  of  x.  From  this  equation 
it  is  at  once  evident  that  all  functions  which  are  at  the  same  time 
integrals  of  /*  =  o  and  Q  =  o  must  also  be  integrals  oi  R  =  o.  For 
convenience  we  will  employ  a  notation  borrowed  from  the  Theory  of 
Numbers  and  write  equation  (3)  in  the  form 

(4)  P~R    mod  Q. 


364  LINEAR  DIFFERENTIAL   EQUATIONS. 

In  this  congruence  P,  Q,  and  7?  denote  differential  expressions  in 
which  the  coefficient  of  the  highest  derivative  is  unity,  and,  denoting 
by  n,  m,  A  the  orders  of  P,  Q,  and  R  respectively,  we  have  the 
inequalities  «  ^  in^-X. 

Suppose  that  all  the  integrals  of  ^  =  o  are  also  integrals  of 
P=:0;  then  it  follows  from  (4)  that  they  must  also  be  integrals  of 
i?  =  O ;  but  the  independent  integrals  oi  Q  =  o  are  w  in  number, 
and  the  order  of  7?  =  O  is  A,  which  is  less  than  w,  and  as  R  =  o  can 
only  have  A  independent  integrals,  it  follows  in  this  case  that  R  is 
identically  zero.  Therefore  :  //  tJie  differential  equation  /*  =  o  has 
among  its  integrals  all  of  the  ijitegrals  of  Q  ^^  o,  theti  P  caji  be  put  in 
the  form 


^r 


P  =  o    mod  Q. 


Again,  suppose  ^  =  o  to  be  an  irreducible  equation,  and  suppose 
the  equation  P  =  o  has  an  integral  Y  which  is  also  an  integral  of 
Q  =.0;  then  the  order  of  Q  cannot  be  higher  than  that  of  P;  if  then 

P=R     mod  Q, 

Fmust  also  be  an  integral  of  i?  =  o;  but  the  irreducible  equation 
Q  =z  O  can  have  no  integral  in  common  with  an  equation  of  lower 
order,  so  that  R  must  be  identically  zero ;  that  is,  we  must  have 

P=  o     mod  Q. 

It  follows  therefore  at  once  that  If  a  linear  differential  equation 
has  among  its  integrals  one  zvJiich  is  also  an  integral  of  an  irreducible 
linear  differential  equation,  tJicn  all  the  integrals  of  the  latter  equation 
.are  integrals  of  the  first. 

Suppose  we  have  two  equations  P  =  o  of  order  n  and  Z',  =  o  of 
•order  ;/, ;  the  integrals,  if  any,  which  satisfy  these  two  equations  will 
be  integrals  of  a  third  equation  which   can   be   found   by  a  process 


IRREDUCIBLE  EQUATIONS.  365-, 

quite  analogous  to  that  for  finding  the  greatest  common  divisor.. 
We  have,  viz., 

P~P^     modP,, 

P,~P,     modP,, 

Pi_,  =  Pi+,  mod  Pi. 
Denoting  by  tik  the  order  of  Pj,,  we  must  have 
n  =  n,  >  «,  >  ;Z3  >  .  .  .  , 

and  Hk  must  vanish  at  the  latest  when  k  =  i  -)-  n^.  Suppose 
«,_,_i  =  o  but  iii  not  zero ;  then  /*,_,_,  either  reduces  to  merely  y  or  is. 
zero.  In  the  first  case  P  =  o  and  P^  =  o  have  no  integral  in  com- 
mon except  jj/=:o;  that  is,  they  have  no  integral  in  common.  In 
the  second  case  the  integrals  common  to  Z'  =  O  and  P^  =  o  are 
integrals  of  P^  =  o,  and  all  the  integrals  of  P^  =  o  are  integrals  of 
P  =  o  and  Pj  =  o.  If,  therefore,  a  linear  differential  equation  is 
reducible,  there  exists  a  linear  differential  equation  of  lower  order  all 
of  whose  integrals  are  also  integrals  of  the  given  equation. 

If  P  =  o  is  a  reducible  linear  differential  equation,  and  ^  =  o  is 
a  linear  differential  equation  of  lower  order  all  of  whose  integrals 
are  integrals  of  P  =  o,  then,  as  we  have  seen,  we  have 

/'=o     mod  Q\ 

that  is,  the  left-hand  member  of  a  reducible  linear  differential  equa- 
tion is  of  the  form 

r.      d'Q    ^       d'  'Q 

We  will  revert  now  to  the  original  definition  of  P  and  Q  as  operators,, 
li     and  add  the  operator  R  defined  by 

i  ^  =  -d?-^''^d^^+  •  •  •  +^'- 


366  LINEAR  DIFFERENTIAL  EQUATIONS. 

Instead  of  the  congruence 

Peo    mod  Q 

we  can  now  write 

Py  =  R{Qy). 

We  recall  that  the  orders  of  P,  Q,  R  respectively  are  n,  m,  /,  and 
n  =  ;«  -|-  /. 

Suppose  za  to  be  the  general  integral  of  the  equation  Rj'  =  o, 
and  V  an  integral  of  0/  =  w ;  then  v  is  also  an  integral  of  Py  =  o, 
and,  from  what  has  been  assumed  concerning  this  equation,  is  a 
regular  integral.  The  function  zu  =  Qv  is  then  also  a  regular  func- 
tion, and  consequently  Ry  =  o  is  an  equation  of  the  same  form  as 
Py  =  o.  The  differential  equation  Qy  =  w  containing  the  /  arbi- 
trary constants  belonging  to  the  function  zv  is  therefore  an  integral 
equation  for  the  given  reducible  equation  Py  =  o,  and  this  last  equa- 
tion is  deducible  from  Qy  =  zv  by  differentiation  and  elimination  of 
the  /  arbitrary  constants.  If  then  a  linear  differential  equation  of 
order  n  has  among  its  integrals  all  the  integrals  of  an  equation 
Qy  ^  o  of  order  in,  each  of  the  integrals  of  the  given  equation  sat- 
isfies a  differential  equation  Qy  =  tc,  in  which  zv  is  an  integral  of  a 
■determinate  equation  of  order  7i  —  in.  Conversely,  a  linear  differ- 
ential equation  is  reducible  when  it  has  for  an  integral  a  differential 
equation  of  the  form  Qy  =  zv.  This  form  of  the  integral  equation 
is  the  characteristic  property  of  reducible  linear  differential  equations. 

So  far  we  have  only  explicitly  considered  equations  of  the  type 
studied  by  Fuchs  in  his  first  two  memoirs,  viz.,  equations  with  uni- 
form coefficients,  a  finite  number  of  critical  points,  and  having  only 
regular  integrals.  We  may  consider  more  generally  linear  differen- 
tial equations  of  which  we  assert  merely  that  they  have  uniform 
coefficients.  An  irreducible  equation  of  this  sort  is  defined  as  one 
which  has  no  integrals  in  common  with  a  differential  equation  of 
lower  order  having  also  uniform  coefificients.  The  theorems  already 
proved  can  be  readily  seen  to  hold  for  this  more  extended  class  of 
equations ;  but  as  the  whole  matter  will  be  taken  up  presently  from 
a  different  point  of  view,  it  is  not  necessary  to  dwell  longer  on  it 
here.     One  general  theorem,  however,  it  is  desirable  to  give,  viz., 


IRREDUCIBLE   EQUATIONS.  367 

that  if  of  tzvo  distmct  integrals  of  a  linear  differential  equation  one  is 
£qual  to  a  differential  expression  with  uniform  coefficients  of  the  other, 
then  the  differential  equation  is  reducible. 
Suppose 

d"y  d"'-y  , 

where  q^,  q^,  .  .  .  ,  q^  are  uniform  functions  of  x.  Denote  by  j„ 
and  jj/j  two  distinct  integrals  of  the  equation 

d"y  d"-y    , 

where /jj/i,  ...,/„  are  uniform  functions  of  x,  and  suppose 

J:  =  Qy.  • 

If  now,  contrary  to  the  statement  in  the  above  theorem,  Py  =  o 
is  irreducible,  then,  since  one  integral  of  Py  ^  o  is  an  integral  of 
P{Qy)  —  o,  all  of  the  integrals  oi  Py  ^  o  must  be  integrals  of 
Pi^Qy)  =  o  ;  if  then  y  is  an  integral  of  Py  —  o,  Qy  is  also  an  integral — 
that  is,  /^  =  O  will  be  satisfied  by  the  functions 

Jo,     Q}\,     Qb'o,     0>o  •  •  •  ; 

but  as  Py  =  o  can  only  have  «  independent  integrals,  we  must  arrive 
at  a  function,  say  Q^y,  which  is  linearly  expressible  in  terms  of  the 
preceding  ones,  or  we  must  have  an  equation  of  the  form 

«oJo  +  «:5/o+    •    •     .    +^kQ'y  =  0, 

where  ^^  is  not  zero,  and  between  the  functions 

Jo,         GJo,     .     •     •     Q'-'fo 

there  is  no  such  linear  relation  with  constant  coef^cients.  The 
number  h  must,  of  course,  be  greater  than  unity,  since  by  hypothesis 
j/„  and  J, ,  =  Qy^ ,  are  distinct  integrals.    If  /^  <  n,  then  we  have  still 


368  LINEAR  DIFFERENTIAL  EQUATIONS. 

to  find  n  —  k  integrals  which,  together  with  those  written  above,  will 
constitute  a  complete  independent  system.     Write  now 

/(O  =  «o  +  «:^  +  •  .  •  +  akr\ 
and 

•^^^^f^  =  fir)  +  flr)s  +  .  .  .  +  flry-\ 

and  consequently 

/.(^)  =  «.  +  «.''+•  •  •  +  <^k>*~\ 


Assume  further 

Ky  =  A{r)y+flr)Qy  +  .  .  .  J^  f,{r)Q'-y, 
then 

^(Or)  =  fir)Qy  +  .  .  .  +y;.(r)G'> 

We  have,  however, 

flr)Q'y,  =  «,S^>o=  —  a,y,  —  a,Qy,  —  .  .  .  —  «*_,S*->„, 

and  consequently 

J^iQyo)  =  -  ^J'o  +  [Mr)  -  a,-\Qy,  +  .  .  .  +  [/.-.W-^.-JQ^-'A 
=  rUir)y,  +flr)Qy,  +  .  .  .  +/..(r)e*->„]  - /(r)j„. 

If  therefore  r  is  a  root  oi  f{r)  =  o,  we  have 

R{Qy:)  =  rRy,, 

and  consequently  each  integral  of  the  irreducible  equation  Py  =  a 
must  satisfy  the  equation 

R{Qy)  =  rRy. 

Denote  by /,-  the  integral  Q'y^ ,  z  =  i,  2,  .  .  . ,  and  we  see  that 

Ry.  -  ^(S>o)  =  rR{Qy:)  =  r'Ry,,. 


IRREDUCIBLE  EQUATIONS.  369 

and  in  general,  for  a  <,  k, 

Ry^  =  r'Ry,. 

If  now  X  starts  from  any  non-critical  point  and  describes  an 
arbitrary  closed  curve  returning  to  the  initial  point,  y^  will  change 
into  an  expression  of  the  form 

c.y.  +  c,y,  +  .  .  .  +  Ck-,yk-.  +  c^yk  +  •  .  •  ^„-i  j„-i, 

and  Ry^  will  become 

(^0  +  ^i^  +  •  •  •  +  Cu-y'~')Ry,  +  c^Ryk  +  .  .  .  +  c,^,Ry„_,, 

and  consequently  all  the  branches  of  this  function  are  expressible  as 
linear  functions  (constant  coefficient  understood)  of 

It  follows  therefore  that  Ry^  satisfies  a  linear  differential  equation 
with  uniform  coefficients  whose  order  is  at  the  most  n—k-\-  i.  The 
linear  differential  equation  Py  z=  o  has  therefore  an  integral 

Ry.  =A{r)y,  -\-flr)y,  +  .  .  .  +  flr)y,_, 

in  common  with  a  differential  equation  of  lower  order,  and  in  con- 
sequence cannot  be  irreducible. 

We  will  now  take  up  the  subject  of  reducibility  of  linear  differ- 
ential equations  with  uniform  coefficients  from  the  point  of  view  of 
the  groups  to  which  such  equations  belong.  Suppose  P  =  o  to  be 
an  equation  of  order  n,  and  Q  =  o  an  equation  of  order  ;;/  <  u  all 
of  whose  integrals  are  integrals  of  /^  =  o.  Denote  by  F,  .  .  .  F„,  a 
system  of  fundamental  integrals  of  Q  =  o:  then  any  substitution 
belonging  to  the  group  of  P=o  can  by  hypothesis  only  change 
V,  .  .  .  F,„  into  linear  functions  of  themselves.  Let  j/,„+,  •  -  .  y„  de- 
note the  functions  which  with  F,  .  .  .  F,„  form  a  fundamental  system 
oi  P=o;  then  the  substitutions  of  the  group  of  P  will  change  these 
functions  in  general  into  linear  functions  of  themselves  and  of 
Fj  .  .  .  F,„  ;  that  is,  all  the  substitutions  belonging  to  the  group 
of  P  will  be  of  the  form 

I  l^x   . . .  n, ;  /    ...  /,„ 


370 


LINEAR  DIFFERENTIAL   EQUATIONS. 


where  f 


f„,  are  linear   functions  of    F, 


F. 


J".+i 


F,„  alone,  and 

•  In- 


fm+i  •  •  '  fn  are  linear  functions  of  F, 

Reciprocally,  if  we  can  choose  j,  .  .  .  y^  such  that  the  group  G 
contains  only  substitutions  of  this  form,  then  these  functions  will 
satisfy  the  equation 


y, 


}\ 


}'n 


dy_^ 

dx^ 

d"'y 
' '       dx-^ 

dy. 

d"'y. 

'dx'      '   ' 

dx'" 

dy- 

d'"y„. 

dx'     '  ' 

' '       dx"^ 

=  o, 


where  the  coefficients  of  y  and  its  derivatives  have  for  ratios  only 
uniform  functions.  Suppose  x  to  describe  an  arbitrary  closed  con- 
tour containing  one  or  more   critical   points ;  then  y^  .  .  .  y^  will 


change  into 


^..J,    + 


+  ^.".J« 


nj,     + 


+  c„„„y„ 


and  consequently  all  the  coefficients  of  this  equation  will  be  multi- 
plied by  one  and  the  same  factor,  viz.,  the  determinant 

I  ^0  I 

of  the  substitution  corresponding  to  the  path  described  by  x ;  the 
ratios  of  the  coefficients  are  consequently  unaltered,  and  these  are 
therefore  uniform  functions.  The  question  of  the  determination  of 
the  group  of  a  given  linear  differential  equation  will  not  be  taken 
up  here,  but  we  can  investigate  the  following  question,  viz.:  Having 
given  a  group  G  composed  of  linear  substitutions  S,  S^,  S^ ,  .  .  .  among 
n  variables,  required  to  detervnne  whetJier  or  not  the  linear  differential 
equation  which  has  G  for  its  group  is  satisfied  by  the  integrals  of 
analogous  equations  of  order  lozver  than  n,  and  to  determine  the 
groups  of  these  equations. 


IRREDUCIBLE  EQUATIONS.  3/1 

From  what  precedes  this  can  be  stated  as  follows : 

Required  to  determine  in  all  possible  manners  a  system  of  linear 
Junctions  Y^,  .  .  .  ,  V,„  of  the  variables  y,,  .  .  .  ,  y,t  such  that  each 
of  the  substitutions  S,  S^,  .  .  .  of  which  G  is  composed  shall  change 
Fj ,  .  .  .  ,  F,„  into  linear  functions  of  themselves. 

This  enunciation  may  still  be  slightly  modified,  and  in  order  to 
do  so  it  is  necessary  to  introduce  a  new  term  which  shall  replace 
the  French  word  faisccau  employed  by  Jordan.  The  author  has 
not  been  able  to  find  any  single  English  word  for  faisceau  which  is 
not  already  employed  for  other  purposes,  or  which  would  be  at 
all  appropriate,  and  therefore  suggests  the  term  function-group. 
We  will  therefore  say  that  a  system  of  functions  forms  a  function- 
group  when  every  linear  combination  of  these  functions  forms  a 
part  of  the  system.  If  Fj,  .  .  .  ,  F„  forms  one  of  the  systems 
of  functions  which  we  have  to  determine  as  above  described, 
then  each  substitution  of  the  group  G  will  transform  the  different 
elements  of  the  function-group  corresponding  to  F^ ,  .  .  .  ,  F,„ 
into  other  elements  of  the  same  function-group.  In  analogy  with 
the  theory  of  groups  of  substitutions,  we  see  that  each  function- 
group  will  contain  a  certain  number  of  linearly  distinct  functions  in 
terms  of  which  every  other  function  in  the  function-group  is  linearly 
expressible.  We  can  say  then  that  the  function-group  is  derived 
from  these  linearly  distinct  functions.  Suppose  now  that  in  the 
function-group  corresponding  to  Fj ,  .  .  .  ,  F^  we  choose  any  other 
m  linearly  distinct  functions;  then  (using  for  brevity  F.  G.  to  denote 
function-group')  every  other  function  of  the  F.  G.  is  linearly  expressi- 
ble, in  terms  of  these  new  m  functions,  and  these  latter  will  further 
form  a  system  possessing  precisely  the  same  properties  as  the  given 
system  F, ,  .  .  .  ,  F,„.  If  now  we  consider  the  F.  G.  instead  of  the 
particular  system  F, ,  .  .  .  ,  F,„ ,  we  shall  have  the  advantage  of  a 
greater  freedom  of  choice  of  the  functions  from  which  the  F.  G.  is 
derived.     Our  problem  may  therefore  be  stated  as  follows : 

Required  to  determine  all  tJie  function-groups  such  that  each  of  the 
transformations  S,  S^,  .  .  .  shall  transform  atiy  element  of  one  into 
.some  other  element  of  the  same  one. 

There  will  of  course  always  be  one  such  F.  G,  formed  by  the 
aggregate  of  the  linear  functions  of  ji ,  .  .  .  ,  j„ .  If  there  is  only 
one  the  group  G  will  be  said  to  be  prime,  and  the  corresponding 


372  LINEAR  DIFFERENTIAL  EQUATIONS. 

differential  equation  will  be  irreducible.  If,  on  the  contrary,  there' 
should  be  more  than  one,  then  to  each  of  the  F.  G.'s  derived  from 
the  linearly  distinct  functions  F, ,  .  .  .  ,  F,„  there  will  correspond 
a  redueed  differential  equation  whose  group  will  be  formed  by  the 
changes  which  the  substitutions  of  G  impose  upon  F, ,  .  .  .   ,  F„, . 

We  have  now  first  to  indicate  a  means  of  ascertaining  whether 
the  group  G  is  or  is  not  prime,  and  in  the  latter  case  to  determine. 
a  function-group  containing  less  than  n  linearly  distinct  elements. 
This  question  will  be  taken  up  in  Chapter  XI.  All  that  immedi- 
ately precedes  concerning  the  group  of  an  equation,  etc.,  is  taken 
directly  from  a  memoir  by  Jordan.*  Chapter  XI.  is  also  taken  from 
this  memoir,  the  only  one  that  the  author  has  knowledge  of  which 
deals  directly  with  the  subject. 

*  Al^moire  sur  une  application  de  la  theorie  des  substitutions  a  T^tude  des  ^quationss 
diff&entielles  lin/aires  :     Bull,  de  la  Society  Mathfematique  de  France,  t.  2,  p.  lOO.. 


L 


CHAPTER   IX. 

LINEAR  DIFFERENTIAL  EQUATIONS  SOME  OF  WHOSE   INTEGRALS 

ARE   REGULAR. 

We  will  now  resume  the  direct  study  of  the  integrals  of  linear 
-differential  equations  with  uniform  coefficients,  and  will  treat  par- 
ticularly the  cases  where  not  all  of  the  integrals  are  regular,  and  show 
how  to  determine  the  number  of  such  integrals  in  each  case.  The 
method  employed  is  due  to  Frobenius  and  in  part  to  Thome.  The 
equation  is 

dy         d"-y        d"-y 

the  critical  points  of  the  coefficients  /,»•••>/«  ^^  to  be  poles, 
and  therefore  in  the  development  of  any  />  in,  say,  the  region  of 
;r  =  o  we  shall  have  only  a  finite  number  of  negative  powers  of  x, 
and  so  for  this  region  each  /  may  be  written  in  the  form 

r  ^  nix) 

where  U  (x)  contains  only  positive  powers  of  x.  Denote  by  go^  the 
degree  of  x  in  the  denominator  of  /j ,  by  go^  the  degree  of  x  in  the 
denominator  of  />^,  etc.  We  will  for  convenience  call  go-  the  order 
of  the  coefficient  /, .     Form  now  the  series  of  numbers  (co^  =  o) 

(3)  ^0  +  ^^>        &9,  +  «  —    I,        Qj^-\-n—  2,       .    .    .    ,        GO,,  , 

which  denote  respectively  by  il^ ,  jQ^ ,  /2, ,  .  .  ,  ,  D,,,.  Let  £■  be  the 
greatest  of  these  numbers ;  obviously,  there  may  be  more  than  one 
/2  equal  to  ^.  Supposing,  however,  that  these  numbers  are  arranged 
in  ascending  order  of  their  subscripts, 

<4)  n,,   n,,   n,,    .  .  .  ,  n„, 

373 


374  LINEAR  DIFFERENTIAL  EQUATIONS. 

we  shall  consider  in  particular  the  first  il,  counting  from  the  left^ 
which  is  equal  to  g.     Suppose  this  is  f2,- ;  then 

(5)  A  =  oOi-Yn  —  i,  =g; 

Jdi  will  be  called  the  cJiaractcristic  index  of  the  given  differential 
equation.  This  notion  of  the  "characteristic  index"  is  due  to 
Thome. 

In  the  case  studied  already,  i.e.  where  all  the  integrals  are  reg- 
ular, it  is  obvious  that  we  have  as  the  greatest  values  of  gj,  ,  .  .  .  ,  Oi?„, 
GiJj  =:  I,  cbJj  ^  2,  .  .  .  ,  and  the  numbers  il„ ,  /2j  ,  .  .  .  ,  £!„  have  n 
for  their  maximum  value,  and  so  /2„  ^  g  ^^  n.  The  theorem  proved 
by  Fuchs  in  this  case  may  now  be  stated  in  the  form  : 

/;/  order  that  the  differential  equation  P  =  o  shall  have  all  of  its- 
hitegrals  regular  in  the  region  ofx^=o  it  is  necessary  that  its  coeffi- 
cients contain  in  their  developments  only  a  finite  number  of  negative- 
powers  of  X,  and,  further,  that  its  characteristic  index  shall  be  zero. 

And  the  converse  theorem  is  : 

If  the  coefficients  ofP^o  contain  only  a  finite  number  of  negative 
powers  of  x,  and  if  further,  the  characteristic  index  of  the  equation  is 
zero,  then  in  the  region  of  x  =  o  all  the  integrals  of  the  equation  are 
regular. 

These  integrals  are,  as  we  know,  each  of  the  form 

^■'  [00  +  0,  log  -1'  +  •   •  •  +  0a  log'^^i'], 

where  the  functions  0  arc  holomorphic  in  the  region  of  ;f  =  o,  and 
therefore  contain  in  their  developments  only  positive  powers  of  x\ 
and  further,  (p^,  .  .  .  ,  0„  do  not  all  vanish  for  x  =  o.  The  expo- 
nents p^,  p^,  '  '  ■  ,  p„  are  roots  of  a  certain  algebraic  equation — the 
indicial  equation. 

The  method  of  arriving  at  this  equation  may  be  briefly  recalled. 
The  differential  equation  is  of  the  form 

^^^  dx^'^     X     dx''-'  ^   '       '   ^     X"     -^  ' 

where  the  functions  U  contain  only  positive  powers  of  x.  Now 
make  J  =  x^,  and  substitute  in  this  equation  ;  we  have 

(7)     xp-"[p{p-  1)  .  .  .  (p-n-\-i) 

+  n,{x)p{p  -  I)  ...  (p  -  «  -  2)+  ...  +  77„(x)]- 


EQUATIONS  SOME   OF    WHOSE  INTEGRALS  ARE  REGULAR.     375 

Multiply  this  result  by  x~^,  and  equate  to  zero  the  coefficient  of 
;r"",  and  we  have  the  indicial  equation 

(8)  p(p_l)    ...    (p -;.+  !) 

+  77,(0)p(p-  I)   .   .   .   (p-;2+2)+   .   .   .   +77„(o)=o. 

The  following  theorems  may  be  easily  proved,  and  the  reader  is 
advised  to  consult  Thome's  memoirs  in  vols.  74  and  75  of  Crelle 
concerning  them.  They  are  given  here  without  proof,  partly  to 
economize  space  but  mainly  because  they  are  in  great  part  included 
in  the  more  general  theorems  which  follow  and  of  which  complete 
proofs  are  given. 

When  the  coefficients  p^,  p^,  .  .  .  , ps  of  the  differential  equation 
P  z=  o  contain  in  their  developments  only  a  finite  nimiber  of  negative 
pozvers  of  x,  and  if  the  equation  has  at  least  n  —  s  regular  linearly  in- 
dependent integrals,  the  remaining  coefficients,  p,^^  .  .  .  /„,  zuill  contain 
in  their  developments  only  a  fi^iite  number  of  negative  pozvers  of  x,  and 
the  characteristic  index  of  the  equation  zvill  be  at  most  eqtial  to  s. 

As  consequences  of  this  we  have  : 

1.  If  the  s  —  I  first  coefficients  /,  .  .  .  p^_^  of  P  =  o  contain  in 
their  developments  only  a  limited  number  of  negative  pozvers  of  x,  and 
if  ps  does  not  satisfy  this  condition,  then  the  equation  has  at  most  n  —  s 
regular  linearly  independent  integrals. 

2.  If  all  the  coefficients  /,  .  .  .  /„  contain  only  a  finite  number 
of  negative  pozvers  of  x,  the  equation  has  at  most  n  —  i  regular  linearly 
independent  integrals  zvherei  is  the  characteristic  ijtdex  of  the  equation. 

If  z  =  o  (Fuchs's  case),  we  know  that  the  equation  always  has  n 
regular  integrals  ;  if,  however,  i  >  o,  it  may  be  that  the  equation  will 
not  have  n  —  i  such  integrals.     For  example  :  Write 

and  consider  the  two  differential  equations 

(9)  Y=o, 
(9)'                                             F+/^  =  o, 

where  h  is  an  arbitrary  function  of  x. 


Z^^  LINEAR  DIFFERENTIAL   EQUATIONS. 

Let  US  form  the  equation 


^-)  ^^-^Z.  =  ^^ 


dY     ,^dh 
h 

that  is, 

/   X  ^^y  I     ^y  , 


where 


dXogJi  dk         dXogk 


All  the  integrals  of  (9)  and  (9)'  satisfy  (4) ;  and  conversely,  as  (10) 
gives  by  integration 

Y=Ch 

y 

if  J/  is  a  solution  of   (11)  or  if  y  verifies    equation  (9),  —  -^  will 

verify  equation  (9)'.  Therefore,  if  (9)  and  (9)'  have  no  solutions  pre- 
senting the  character  of  regular  integrals,  equation  (ii)  will  have  no 
regular  integrals. 

Let  /,  and  h  be  taken  arbitrarily, 


whence 


/.  =   — 4  '       ^^   =  ^^ 


-^  =  A  +  -; 

X  X 


that  is,  for  ;f  =  o,  k  is  infinite  of  the  order  4.  This  order  being 
superior  to  i,  equation  (9)  has  no  regular  integral.  Equation  (9)'  has 
now  no  solution  of  the  nature  of  regular  integrals,  for  its  general 
integral  is 

(12)  y  =  e-  f '•"%€'  - / he^^'^'^dx) ; 

or,  replacing  h  and  k  by  their  values  and  effecting  the  integration 
within  the  brackets, 

(13)  y  =  e-f''^\&{x)^C^ogx\ 


EQUATIONS  SOME   OF    WHOSE  INTEGRALS  ARE  REGULAR.     ^77 

C^  being  a  constant,  and  the  function  S{x)  containing  in  its  develop- 
ment an  unlimited  number  of  powers  of  x"^,  as  well  as  e'^''^". 

Therefore,  none  of  the  integrals  of  (9)  and  (9)'  being  regular,  equa- 
tion (11)  has  no  regular  integrals.     It  is  of  the  form 

where  f{x)  is  a  holomorphic  function  in  the  region  of  the  point 
zero,  since 

f{x)   =-2X^-    5. 

The  characteristic  index  is  i,  and  nevertheless  the  equation,  by  what 
precedes,  has  no  regular  integral. 

The  preceding  method  admits  of  generalization  and  enables  us 
to  form  differential  equations  of  any  degree  with  coefficients  infinite 
of  finite  order  for  ;tr  =  o  and  not  possessing  w  —  i  linearly  inde- 
pendent regular  integrals,  i  being  their  characteristic  index. 

Start  now  with  the  equation 

in  which  the  uniform  coefficients  p  contain  only  a  finite  number  of 
negative  powers  of  x  {i.e.,  we  will  only  consider  the  region  of  the 
critical  point  x  =  o).  If  we  substitute  for  y  the  value  y  =  x^,  we 
obtain  a  function  of  x  and  p,  viz.,  P{xf),  which  Frobenius  has  called 
the  characteristic  function  of  the  differential  equation  /*  =  o,  or  of 
the  differential  quantic  P.     We  find  at  once 

(,5)  Pi.')  =  ,,pP-)  ■■>->>  +  ■)  ^^_  .(.-.)  .^.  (.-.+.) 

We  see  at  once  that  the  product  x-pP{xp)  can  be  developed  in  inte- 
ger powers  of  x,  and  that  the  development  will  only  contain  a  finite 
number  of  negative  powers  of  x;  the  coefficients  in  the  development 


378 


LINEAR  DIFFERENTIAL  EQUATIONS. 


will  contain  only  positive  powers  of  p,  and  p"  is  the  highest  power 
that  can  occur.  When  the  differential  quantic  P  is  given,  its 
characteristic  function  P{x^)  is  of  course  known.  Suppose,  how- 
ever, a  characteristic  function  xPf{x,  p)  to  be  given  where /(;r,  p)  is 
an  integral  function  of  p  whose  coefficients  are  functions  of  x  : 
what  is  the  corresponding  differential  quantic  ?     Write 


(i6) 
and 

(17) 


f{x,  p+  I)  -  f{x,  p)  =  Apf{x,  p) 


U,= 


I  .  2 


We  know  now  that  the  integral  function  /{x,  p)  of  p  can  be  placed 
in  one  way,  and  in  one  way  only,  in  the  form 


(i8)    /{x,  p)=  Unp{p-  I)  . 

+  Un-rP{p—   I) 

it  follows  then  that 
(19)     xPf{x,  p) 


{p  —  n—  i) 
.  (p  -n-\-2)-\- 


=  XP 


ILx 


„p{p  -i)  .  .  .  (p-?/+i) 


+ 


-|-^.P+C/„ 


-{-U,x'--+[/, 


is  the  characteristic  function  of  the  differential  quantic 
d"v  d"~^v  dv 

^•-"-2^+  ^-^"-  5S^  +  •  •  •  +  ^■^■:^-  +  ^•^- 

We  know  that  the  product 

p(p-i)  .  .  .  (p-«+  i) 


dx 


(20)     x-pP{xP) 


+  /. 


p(p-  I)  .  .  .  {p-n-^2) 


+ 


+  /«-!-  +  A 


can  be  developed  in  a  series  of  ascending  powers  of  x,  containing 
only  a  finite  number  of  negative  powers  and  in  which  the  coefficients 
are  integral  functions  of  p  of  degrees  at  most  =  n.     We  wish  now 


I 


EQUATIONS  SOME   OF    WHOSE  INTEGRALS   ARE  REGULAR.     379^ 

to  determine  the  first  term  of  this  series.  The  exponents  of  ;rin  the 
denominators  of  the  expansion  of  x~^P{xp^  are  obviously  the  num- 
bers n„  ,  ill  ,  .  .  .  ,  /2„  above  defined.  If  g  is  the  greatest  of  the 
numbers  fl,  the  first  term  of  the  series  will  clearly  be  of  the  form 
r~'(  \ 
— ^,  where  6^(p)  is  an  integral  function  of  p  and  does  not  contain  x. 

If  we  denote  by  y  the  degree  of  G{^p),  we  have  obviously 

(21)  y  ^  n  —  i, 

where  i  is  the  characteristic  index  of  the  equation,  i.e.,  i  is  the  index 
of  the  first  jQ  which  is  =  g.  In  the  particular  case  of  z  =  o  the 
equation  G'(p)  =:  o  is  the  indicial  equation  already  defined  in  the 
case  where  all  the  integrals  of  P  =  o  are  regular.  In  the  case  of 
/  >  o  we  can  now  generalize  this  notion  of  the  indicial  equation  and 
say  that  Gi^p)  =  o  is  the  indicial  equation  of  our  present  differential 
equation  P=  o  or  of  the  differential  quantic  P.  The  function  G{^p) 
will  be  called  the  indicial  function. 

Thus  to  obtain  the  indicial  function  of  a  difTerential  quantic, 
we  form  its  characteristic  function  and,  after  multiplying  by  ^—p, 
develop  the  product  in  ascending  powers  of  x\  the  coefficient  of  the 
first  term  is  the  indicial  function.  It  is  to  be  noted  that,  knowing  _^, 
it  is  sufficient  to  multiply  the  characteristic  function  by  x^-^  and 
then  make  ;tr  =  o  ;  thus 

is  the  indicial  function  of  P{}'). 

Some  properties  of  the  indicial  function  of  the  differential  quan- 
tic Pwill  now  be  given. 

First:    if /„  is  identically  zero,  the  characteristic    function  and 

consequently  the  indicial  function  is  divisible  by  p.     Effecting  this 

division   and  changing  p  into  p  -|-  i  in  the  quotient,  we  obtain,  by 

equating  it  to  zero,  the  indicial  equation  of  the  differential  equation 

dy 
of  order  n  —  i  obtained  by  taking  —  for  the  unknown  variable. 

Second  :  if  in  the  equation  P  =  o  we  put 

(22)  y  =.  xPt'W, 


38o 


LINEAR  DIFFERENTIAL  EQUATLONS. 


the  indicial  equation  of  the  differential  equation  in  w  thus  obtained 
will  have  for  its  roots  those  of  the  indicial  equation  of  P  =  o  di- 
minished by  P(, .  For  the  characteristic  function  of /*(;irPo2X')  =  o  is 
P{xfo'^f).  It  is  therefore  deduced  from  the  characteristic  function 
of  the  equation  in  y,  by  changing  p  to  p  -j-  p„ ;  and,  consequently, 
the  same  is  true  of  the  indicial  equations. 

Finally :  if  in  the  equation  P  =  o  we  place 


<23) 


y  =^  rp{x)'W, 


tp{x)  being  a  holomorphic  function  in  the  region  of  the  point  zero 
and  not  vanishing  for  ;ir  =  o,  the  indicial  equation  of  the  equation  in 
w  thus  obtained  will  be  the  same  as  that  of  the  equation  in  y.  For 
it  is  easily  seen  that,  the  equation  in  w  being  of  the  form 

its  characteristic  function  is  the  sum  of  two  terms.     The  first  term, 
-p{p-  i)  .  .  .  (p-«  +  I) 


(25)     xo 


X" 


p{p-i)  .  .  .  {p-n+2) 
+  /. Zi^. h  •  •  •  +/. 


is  the  characteristic  function  of  the  equation  in  y,  and  in  the  second 
term, 

-p{p—  i)  .  .  .  {0—  n-]-i) 


(26)     XP 


+  ^. 


p(p  —  i)  .   .  .  (p  -  «  +  2) 


.     .     .     +Pn~\, 


the  highest  exponent  of  x  in  the  denominator  is  inferior  to  the  high- 
est power,  g,  in  the  denominator  of  the  first  term.  Whence  it  follows 
that  upon  multiplying  by;ir'^-''  and  then  making  ^  =  o  to  obtain 
the  indicial  function  of  the  equation  in  w,  the  second  term  will 
vanish  and  the  result  will  be  the  same  as  if  the  operation  had   been 


EQUATIONS   SOME    OF    WHOSE   INTEGRALS  ARE   REGULAR.     38 1: 

performed  upon  the  first  term  alone,  that  is,  it  will  be  the  indicial 
function  of  the  equation  my. 

It  follows  from  these  propositions,  p^,  p^,  .  .  .  ,  Py  being  the  y 
roots  of  the  indicial  equation  of  P: 

1.  That  if  in  P  =  O  we  put 

(27)  y  =  y,w, 

where  J,  is  an  integral  of  the  form  xPoip{x),  ^'(,r)  being  holomorphic 
in  the  region  of  the  point  o  and  not  zero  for  x  =  o,  the  homogeneous 
differential  equation  in  w  thus  obtained  will  have  an  indicial  equa- 
tion whose  roots  are 

Pi  —  P.,      P^  —  Pc      '   '   •   ,     Py—  Po, 

and,  this  equation  being  divisible  by  p,  one  of  the  quantities  p, ,  />, , 
.  .  .  ,  Py  must  be  equal  to  p^ ;  let  p^  =  p„. 

2.  If  in  the  equation  in  za  we  now  put 

w  =^fzdx^ 

the  indicial  function  of  the  equation  in  z  will  be  of  degree  y  —  i>^ 
*  and  will  have  for  roots 

P2  — Po—  I.     Pa  — Po—  I.     ...,     Py  —  p,—  \. 

3.  Consequently,  if  in  /*  =  o  we  make  the  substitution 

y  =  y,fzdx, 

the  equation  in  z,  of  the  order  n  —  i,  thus  obtained  will  have  for  in- 
dicial equation  the  equation  of  degree  y  —  i  which  admits  the  roots 

p2  —  Po  —  I,     •  .  .  ,     py  —  p^—  I. 

The  simple  relation 

(28)  i-\-  y  =  7t 

between  the  order  of  the  differential  equation,  its  characteristic 
index,  and  the  degree  of  the  indicial  equation,  enables  us  to  replace 
the  notion  of  the  characteristic  index  by  the  more  rational  consid- 
eration   of   the  indicial  equation.     The  characteristic  index  of   an. 


382 


LINEAR  DIFFERENTIAL  EQUATIONS. 


equation  is  merely  the  difference  between  the  degree  of  the  equation 
and  that  of  its  indicial  equation.  Whence  it  is  evident  that  all 
propositions  concerning  the  characteristic  index  and,  consequently, 
relating  to  equations  whose  coef^cients  are  infinite  of  a  finite 
order  for  x  =■  o  can  be  expressed  by  aid  of  the  indicial  equation. 
Thus : 

The  number  of  linearly  independent  regular  integrals  of  the  equa- 
tion P  ^=  o  is  at  most  equal  to  the  degree  of  its  indicial  equation. 

To  obtain  a  clearer  idea  of  the  characteristic  and  indicial  func- 
tions of  P  =2  o,  the  equation  may  be  put  in  a  certain  normal  form. 
Thus  we  may  write 


(29)  P{y)=-^x 


dx"^  X"- 


X 


d"-y 

.n-i  -^ 


dx"- 


I     /, 

Reducing  the  fractions  — -  ,  --— 

X         X    ' 


A-: 


,  p„    to    the   least  com- 


mon denominator  x^  and  multiplying  by  x^ ,  the  equation  becomes 


dy  d"-'y 


dy 


where  the  functions 


contain  only  positive  powers  of  x  and  do  not  all  vanish  for  x  =  o. 

This  form  of  the  first  member  of  a  linear  differential  equation 
will  be  called  the  nor/nal  form.  The  characteristic  function  of  the 
differential  quantic  T  is 


(31)     T{x') 


'-xp{t,p{p-  I) 
+  /,p(p-  I)  . 


.  (p  -  m  +  i) 

(p  _  ;/,  +  2)  + 


+  ^«-i/>  +  4]; 


I 


hence  the  product  x-fT{xP)  contains  only  positive  powers  of  x  and 
does  not  vanish  for  x  =^0]  its  constant  term  is  the  indicial  function. 
Conversely,  it  is  easy  to  see  that  if  a  linear  differential  equation 
has  a  characteristic  function  fulfilling  these  conditions,  it  is  in  the 
normal  form.  Hence  an  examination  of  the  characteristic  function 
is  suf^cient  to  determine  whether  the  differential  equation  is  or  is 
not  in  the  normal  form. 


EQUATIONS  SOME   OF    WHOSE  INTEGRALS  ARE  REGULAR.      383 

Our  next  step  will  be  to  define  a  composite  differential  quantic 
and  prove  one  of  its  important  properties. 
In  the  differential  expression 

d'^v  d°-~^v  dv 

T-)    ^(/)=^.z?+^.s^+  ■■■  +  ^-.£+^.j' 

the  letter^  will  be  considered  a  symbol  of  operation  such  that  A{}') 
indicates  the  definite  operation  to  be  performed  upon  /: 


(33)  ^W  =  K;5i+^.S;;n+-..+^.l/. 

Also,  A{B),  or  simply  AB,  will  indicate  that  the  same  operation  is 
to  be  performed  upon  B.  If  then  ^  is  a  differential  quantic,  AB 
will  also  represent  a  differential  quantic  C,  and  we  shall  say  that 
the  quantic  AB  =  C  is  composed  of  the  qualities  A  and  B  taken  in 
that  order.  The  same  definition  holds  for  a  quantic  composed  of 
more  than  two  quantics. 

If  the  coefficients  of  the  component  differential  quantics  con- 
tain only  a  limited  number  of  negative  powers  of  x  as  is  here  sup- 
posed, it  is  clear  that  it  is  the  same  with  the  composite  quantics. 

Let  AB  =  C,  and  let 


(34) 


A  (a'p)  =  x^h{x,  p)  =  ^//^(pU'P  +  ^, 
B{x'>)  =  xPh{x,  p)  =  2,k,{p)xi'  +  \ 
L  C{xf)  =  xH{x,  p)  =  ^J^{p)x'>  +  ^, 


be  the  characteristic  functions  of  the  quantics  A,  B,  and  C. 
We  have 

135)       C{x9)  =  AB{xo)  =  A  [^,/&,(p>p  +  '']  =  :2,k,{p)A{x'>  +  ^) ; 

that  is, 

(36)  2  k  4  {f>)x^  =  2^:2M,p  +  y)k^{p)x'^  + " . 

From  this  equality  it  follows  that  if  A  and  B  have  the  normal 
form,  since  in  that  case  h^{p)  and  k^{p)  vanish  for  negative  values  of 
yu  and  V  but  not  for  the  value  zero,  2A.4(p)-^^will  contain  only  posi- 
tive powers  of  x,  and  will  not  vanish  for  x  =  o.  Therefore,  the  char- 
acteristic function  of  C  divided  by  xp  fulfilling  these  conditions,  C 


384 


LINEAR  DIFFERENTIAL  EQUATIONS. 


will  itself  have  the  normal  form.  Moreover,  making  ;ir  =  o  in  the 
same  equation,  we  have  the  following  simple  relation  between  the 
indicial  functions  of  A,  B,  and  C : 


(37) 


4(p)  =  K{p)klp)- 


We  may  therefore  state  the  following  proposition : 

If  a  differential  qjiantic  is  composed  of  sevo'al  differential  qiian- 
tics  each  of  wJiich  is  in  the  normal  form,  it  has  itself  the  normal 
form,  and  its  indicial  function  is  the  product  of  the  ifidicial  functions 
of  the  component  quantics. 

The  degree  of  IJ^jS)  is  consequently  the  sum  of  the  degrees  of 
/z„(p)  and  k,{p). 

More  generally,  we  may  remark  that  if  two  of  the  tlvree  differen- 
tial quantics  A,  B,  and  C  have  the  normal  form,  the  third  has  also' 
the  normal  form. 

The  notion  of  the  component  factors  of  a  differential  quantic 
leads  directly  to  that  of  reducibility ;  but  as  this  subject  has  been 
treated  elsewhere,  it  will  not  be  resumed  in  this  connection. 

Resuming  now  the  question  of  regular  integrals,  we  see  at  once 
that  if  the  equation  P^  o  has  a  regular  integral,  then  P=  o  is  a  re- 
ducible equation.  For,  if  P  =  o  has  a  regular  integral,  it  necessarily 
has  one  of  the  form  xc  rp{x),  where  ij^{x)  is  a  uniform  function  which, 
does  not  vanish  for  x  =^  o.     Now  obviously  the  function 


(38) 


is  an 


y.  =  X'  i{x) 


integral  of  the  equation  of  the  first  order 


(39) 
where 

(40) 


dy    ,  Pix) 


dx 


X 


-  y  —  o. 


X  "^  iix)  ' 


Pix)  containing  only  positive  powers  of   x.      Therefore  P=o  is 
reducible  and  has  among;  its  integrals  all  those  of 


(41) 


B.= 


d^      P^) 
dx^    X 


y  =  o. 


EQUATIONS    SOME   OF    WHOSE  INTEGRALS  ARE   REGULAR.     385 

We  can  now  place  P  =  o  in  the  form  Q^B^  =  o,  where  Q^  is  of  order 
71—1.     If,  again,  Q^  =:  o  has  a  regular  integral,  we  can  write 

(42)  Q.  =  Q.B,, 

where  Q^  is  of  order  n  —  2.     Continuing  this  process,  we  have  finally 

(43)  P=QD, 

where  D  =^  B^B^^^  .  .  .  B^, 

and  where  each  B  equated  to  zero  gives  a  differential  equation  of 
the  first  order  having  only  regular  integrals,  and  ^  =  o  is  a  differen- 
tial equation  of  order  n  —  (3  which  has  no  regular  integral.  If  P  is 
in  the  normal  form,  the  same  thing  may  obviously  be  supposed  of 
Q  and  D.  The  regular  integrals  of  /*  =  o  being  now  those  of  /)  =  o, 
suppose  P  =  o  has  all  of  its  integrals  regular ;  it  follows  at  once  that 
/3,  the  degree  of  Z>  =  o,  is  equal  to  Ji  (/5  =  n),  and  so  that  P  =  D. 
We  have  then  the  theorem  : 

{a)  If  the  equation  P  =  O  has  all  of  its  integrals  regular^  it  can  be 
composed  in  one  zvay,  and  one  way  only.,  of  differential  egiiations  of  the 
first  order  having  each  only  regular  iiitegrals. 

The  converse  of  this  is  : 

(/?)  If  the  differential  equation  P  ^^o  is  composed  in  07ie  way.,  and 
one  way  only,  of  equations  of  the  first  order  each  of  zvhich  has  only 
regtilar  integrals,  it  will  have  all  of  its  integrals  regular.  The  proof 
of  this  will  be  left  as  an  exercise  for  the  reader. 

The  following  theorems  are  immediately  inferrible  from  what 
we  have  just  shown,  viz. : 

(i)  If  P=^  O  admits  of  regular  integrals,  there  exists  an  equation 
Z)  =  o  all  of  whose  integrals  are  regular,  and  these  are  the  regiilar 
hitegrals  of  P  ■=^  o. 

(2)  //"  Z?  =  o  is  the  equation  zvhicJi  gives  the  regular  integrals  of 
P  =  O,  and  if  zve  put  P  in  the  composite  form  P  =  QD,  tJien  the  equa- 
tion Q  =:  o  has  no  regular  integrals. 

If  ^  =  O  and  ^  =  o  are  two  equations  having  all  their  integrals 
regular,  then  they  can  be  each  composed  uniquely  of  equations  of 
the  first  order  each  of  which  admits  only  regular  integrals.  The 
equation  AB  =  o  will  then  be  composed  in  the  same  way.  And  so 
for  any  number  of  equations  A=o,  B^=^o,  C=o.  .  .  We  have 


3^6  LINEAR  DIFFERENTIAL  EQUATIONS. 

then  this  generahzation  of  /? :  If  the  equation  P  =  o  is  composed 
uniquely  of  equations  of  lower  orders  which  have  all  their  integrals 
regular,  then/*=:  o  will  have  all  of  its  integrals  regular.  From  what 
precedes,  we  know  that  any  linear  differential  quantic,  say  A,  can  be 
put  in  the  form  A  =  QD  (where,  of  course,  the  order  of  D  may  be 
zero),  in  which  Z?  =  o  has  only  regular  integrals  and  ^  =  o  has  no 
such  integrals.  Suppose  a  new  equation  ^  =  o  to  have  only  regular 
integrals;  then 

(44)  AB=Q{DB). 

Now,  as  already  shown,  the  regular  integrals  of  AB  =  o  are  those 
of  DB  =  o ;  but  by  hypothesis  DB  =  o  has  all  of  its  integrals  reg- 
ular ;  it  follows  then  that  the  number  of  linearly  independent  integrals 
of  AB  =  o  is  equal  to  the  sum  of  the  orders  of  the  differential  equa- 
tions B  =^  o  and  D  =  o,  or,  what  is  the  same  thing,  is  equal  to  the 
sum  of  the  linearly  independent  integrals  oi  A  =  o  and  B  =  o.  It 
follows  from  this  that  if  tJie  equation  B  =.0  has  only  regular  integrals, 
the  equation  AB  ^  o  will  have  exactly  as  many  regular  integrals  as 
the  two  equations  A  =  o  and  B  =^  o  have  combined. 

From  this  we  draw  at  once  the  conclusion  \  If  B  ^=  o  has  only 
regular  integrals,  and  if  AB  =^  o  has  s  of  such  integrals  zvJiich  are 
linearly  independent,  then  A  =  o  will  have  s  —  ft  linearly  independent 
regular  integrals,  ft  denoting  the  order  of  B. 

Before  going  farther  it  will  be  convenient  to  make  a  remark  con- 
cerning a  differential  equation  of  the  first  order.  Suppose  the  equa- 
tion is 

dy 

(45)  ux  —  +  zvy  =  o, 

and  suppose  further  that  it  is  in  the  normal  form ;  then,  of  course, 
u  and  IV  contain  only  positive  powers  of  x  and  do  not  both  vanish 
for  x  =  o.     The  general  integral  of  this  equation  is 

/'  ivdx 

(46)  y  =  e   '^    «^. 

Now  we  know  that  if  u  does  not  vanish  for  x  =0,  y  will  be  a  regu- 
lar integral  of  the  form  xPf{x);  if,  however,  for  x  =  O  we  have 
u  =  o  oi  order  h  (in  which  case  of  course  w  is  not  zero),  then 

(47)  y  =  e-^-  '^-■■^^■■■^-''xo>p{x), 


EQUATIONS    SOME   OF    WHOSE  INTEGRALS  ARE  REGULAR.     387 

\vhich  is  not  a  regular  integral.  The  function  ^\x)  contains  only- 
positive  powers  of  x  and  does  not  vanish  for  x  =  o. 

Suppose  11^  and  w^  to  be  the  values  of  it  and  zv  for  x  ^=  o.  The 
characteristic  function  of  the  considered  equation  is  x^iiip  -f-  w),  and 
the  indicial  function  is 

(48)  u,p  +  iv^ . 

In  the  first  case,  then,  where  ?/„  is  not  zero,  and  consequently  where 
the  equation  has  its  integrals  regular,  the  indicial  function  is  an 
integral  function  of  p  of  the  first  degree  ;  and  in  the  second  case 
where  ?/„  =  o  and  the  equation  has  no  regular  integral  the  indicial 
function  is  a  constant.     The  converse  of  this  is  readily  shown. 

It  is  obvious  from  what  precedes  that  if  the  equation  /*  =  o  has 
all  of  its  integrals  regular,  then  the  degree  of  its  indicial  equation  is 
equal  to  its  order ;  and  also  that  the  number  of  linearly  independent 
regular  integrals  of  P  =  o  is  at  most  equal  to  the  degree  of  its 
indicial  equation.  These  results  have,  however,  been  previously 
arrived  at. 

We  will  now  obtain  the  precise  condition  which  the  differential 
equation  7^=0  must  satisfy  in  order  that  the  number  of  its  linearly 
independent  regular  integrals  may  be  exactly  equal  to  the  degree 
of  its  indicial  function.  We  suppose  the  differential  quanticsP,  Q,  D 
to  be  in  the  normal  form.  If  the  number  of  linearly  independent 
regular  integrals  of  /-"  =  o  is  y,  where  y  is  the  degree  of  the  indicial 
function  of  P,  then  P  can  be  placed  in  the  form 

P=  QD, 

where  Q  is  of  the  order  n  —  y  and  has  a  constant  for  its  indicial 
function.  For,  since  P=^  o  has  y  regular  integrals,  it  can  be  placed 
in  the  form  P  =  QD,  where  Q  is  of  order  u  —  y  and  has  no  regular 
integrals,  and  where  D  is  of  order  y  and  has  all  of  its  integrals  regu- 
lar. The  indicial  function  of  P  is  then  the  product  of  the  indicial 
functions  of  Q  and  D  ;  but  since  D  has  all  of  its  integrals  regular, 
its  indicial  function  is  of  degree  y,  and  this  is  also  the  degree  of  the 
indicial  function  of  P;  hence  the  indicial  function  of  ^  is  a  constant. 
Conversely,  if  the  differential  expression  P  having  an  indicial 
function  of  degree  y  can  be  placed  in  the  form  P  =  QD,  where  Q  is 


388 


LINEAR  DIFFERENTIAL  EQUATIONS. 


a  differential  expression  of  order  n  —  y  having  a  constant  as  its  in- 
dicial  function,  then  /*  =  o  will  have  exactly  y  regular  integrals. 
For,  since  P  is  of  order  ;/  and  Q  of  order  n  —  y,  D  is  of  order  y  ; 
now  the  indicial  function  of  Q  being  of  order  zero,  that  of  D  is  of 
order  y,  and  since  the  indicial  function  of  ^  is  a  constant,  Q  can 
have  no  regular  integrals,  and  so  the  equation  P  ^=.  QD  ^  o  will 
have  for  regular  integrals  those  of  Z'  =  o;  that  is,  /*:=  o  has  ex- 
actly y  regular  integrals.  Therefore,  finally  :  In  order  that  P  =  o 
'  hazdng  an  indicial  function  of  degree  y  shall  have  y  regular  integrals, 
it  must  be  possible  to  place  P  in  the  form  P  ^  QD,  ivhcrc  Q  is  of 
order  n  —  y  and  has  a  constant  for  its  indicial  finction. 

Irregular  integrals  of  the  form  (47)  have  been  called  by  Thom^ 
normal  integrals.  It  is  not  the  intention  here  to  go  at  all  into  the 
theory  of  these  integrals,  indeed  that  theory  is  in  a  very  imperfect 
state  owing  principally  to  the,  in  many  cases,  impossibility  of  ascer- 
taining anything  definite  concerning  the  convergence  of  the  series 
involved.  The  reader  is  referred  to  Thome's  papers  in  Crelle,  but 
particularly  to  the  following  papers'  by  Poincare  :  "  Sur  les  equations 
lindaircs  aux  diffe'rentiellcs  ordinaires  et  aux  differences  finics"  (Amer- 
ican Journal  of  Mathematics,  vol.  7,  pp.  208-258),  "  Sur  les  integrals 
irregulilres  des  Equations  lindaires'  (Acta  Mathematica,  vol.  8,  pp.  295— 
344);  see  also  a  not*  by  Poincar^  in  the  Acta,  vol.  10,  p.  310,  en- 
titled "  Remarques  sur  les  intcgrales  irre'gulieres  des  dquations  linc- 
aires.''  A  Thesis  by  M.  E.  Fabry  (Paris,  1885)  entitled  ''Sur  les  in- 
tc'grales  des  (Equations  diffe'rentiellcs  lindaircs  a  coefficients  rationnels" 
may  also  be  advantageously  consulted.  The  first  paper  by  Poincare 
in  the  Acta  Math,  is  by  far  the  most  important  of  these  references  ; 
the  author  regrets  to  be  unable  to  give  an  account  of  it  here,,  but 
limits  of  space  prevent. 


CHAPTER   X. 

DECOMPOSITION   OF  A   LINEAR   DIFFERENTIAL   EQUATION   INTO 
SYMBOLIC   PRIME   FACTORS. 

The  coefficients /j ,  .  .  .  ,/„  of  the  differential  equation 

are,  with  the  exception  of  certain  isolated  critical  points,  holomof' 
phic  functions  of  ;f  in  a  portion  of  the  plane  which  is  limited  by  a 
simple  {i.e.,  non-crossing)  contour. 
P  can  be  put  in  the  form 

<2)  P  =  A,A,,.,  .  .  .  A,, 

where 

(3)  ^'  =  ^£-'''^ 

is  called  a  symbolic  prime  factor  of  P.  For  convenience  the  word 
*'  symbolic"  may  be  omitted,  as  it  is  always  understood.  We  can  now 
show  that  P  is  decomposable  into  prime  factors.     Let 

(4)  y,  =  v,,    y,  =  vjv.dx,    .  .  .  ,    ;/„  =  vjv^xfv.dx  .  .  .  fvjx 

denote  a  set  of  fundamental  integrals  oi  P  =.  o.  Form  now  the 
system  of  differential  equations 


dv 
<5)  ^i—  dx~  ^'^~^'   1=1,2,  ...  ,  n, 


389 


390  LINEAR  DIFFERENTIAL   EQUATIONS. 

having  for  integrals 


then 
(6) 


V,  ,     v^v, ,     v,v,v, , 


d 


V{V^  .  .  .  v^\ 


tti  =  -^  log  {2\V,    .    .    .    Vi). 


It  is  obvious  now  that  the  composite  expression  A„A„_^  .  .  .  A^. 
is  identically  equal  to  P\  that  is,  the  coefficients  of  the  derivatives  of 
y  of  the  same  order  are  the  same  in  both  expressions,  a  fact  already 
established. 

It  is  easy  to  see  now  that  every  decomposition  of  P  into  prime 
factors  is  obtainable  by  this  process,  each  such  decomposition  corre- 
sponding to  a  chosen  system  of  fundamental  integrals.  We  see  at 
once  an  analogy  between  algebraic  equations  and  linear  differential 
equations.  When  we  know  the  roots  of  the  algebraic  equation,  its 
decomposition  into  prime  factors  enables  us  at  once  to  write  out  the 
equation  ;  so,  knowing  the  decomposition  into  prime  factors  of  a 
differential  equation,  we  are  enabled  to  form  the  differential  equation, 
which  possesses  a  given  system  of  fundamental  integrals.     If 


7,  =  ^^i»    y-.^'^J^^'-^x, 


yn  =  vjz\dx  .  .  .  fv,4x 


are  the  integrals,  the  differential  equation  will  be 

(7)  A,An-.  .  .  .  A,=o; 

where 

d 


,  11. 


It  is  to  be  noticed,  however,  that  while  the  arrangement  of  the 
factors  in  the  algebraic  equation  is  arbitrary  {i.e.,  the  factors  are 
commutative),  the  arrangement  of  the  symbolic  factors  in  the  differ- 
ential equation  is  not  in  general  arbitrary. 

When  we  know  a  system  of  values  of  the  functions  i\  .  .  .  v„  we 
know  also  all  possible  systems  of  fundamental  integrals  oi  P  =^  o, 
and  so  can  in  all  possible  ways  decompose  P  into  its  prime  factors 
by  aid  of  the  formulae 


d 
(9)  ^i  =  ^  log  {v, , 


>>)- 


I,  2, 


n. 


DECOMPOSITION  INTO    SYMBOLIC  PRIME  FACTORS.         39I 

Conversely,  if  we  know  the  coefficients  a  of  the  prime  factors,  we 
can  integrate  P  =  o ;  for  we  have 

Jaxdx  f{ai—ai)dx  f{a„  —  an-i^x 

(10)  z\  =  e         ,     v,=  e  ,    .  .  .  ,    v„=  e  , 
and  a  system  of  fundamental  integrals  of  P  =  o  is  then 

faidx  faidx       f{a2—ai)dx 

(11)  7j=   ^  ,      ^2=   ^  Je  ,       .    .    .. 

If  now  we  wish  to  decompose  the  expression 

into  its  prime  factors,  we  have  one  of  two  things  to  do :  either  cal- 
culate a  system  of  values  of  the  functions  i\,  v^,  .  .  .  ,  v„ ,  or  evalu- 
ate directly  a  system  of  values  of  the  coefficients  a^,  a^,  .  .  .  ,  a„. 
In  the  first  case  we  should  have  first  to  find  a  particular  solution 
7'j  of  the  equation  /'(^,)  =  o,  then  a  particular  solution  v^  of 
P{i\fz^dx)  =  o,  etc.  The  equations  in  ^^ ,  ^-^ ,  .  .  .  ,  z„  will  be  linear 
and  of  orders  n,  7i  —  i,  .   .  .  ,  2,   i  respectively.     In  the  second  case 

we  should  have  to  find  first  a  particular  solution  a^  of  Pi^e''"^';  =  o, 
then  a  particular  solution  a,  of  /?[^-/'""^^y^/(«^-"')^-^^.r]  =  o,   etc. 

The  equations  in  u^,  u^,  .  .  .  ,  u„  will  be  of  orders  n,  n  —  i,  .  .  .  , 
2,  I,  but  will  obviously  not  be  linear. 

The  formulae  giving  the  coefificients/j  .  .  .  p„  of 

d"y  d"-'v 

in  terms  o(  a^,  a^,  .  .  .  ,  a„  are  easily  found.     Suppose 

(12)  P  =  A,A„_,.  .  .  A,, 
and  further  suppose 

(13)  A^A^_,  •  •  •  ^^  =  5^  +  ^^  d^  +  •  •  .  +  gn-.y. 


392 


LINEAR  DIFFERENTIAL  EQUATIONS. 


dy 
In  this  last  equation  replace  y  by  -r- — «i  J,  and  so  form  the  expres- 
sion A„A„^,  .  .  .  A^A^,  or  P{y) ;  we  have  then 

,    ,  „,  s     d"y  ,  ,        .d"-y     r  ^^.         -]^»-> 


+  ...+ 
from  which  we  find 

(15)   Px  =  q.  —  a,, 
P.  =  g.-  («  -  0 


dx"- 


-  .  .  .  -  q,. 


da^ 

'dx 


dx"- 


—  ^»-i«, 


j; 


da^ 

llx 


^i«i 


P^ 


{n  —  i){n  —  2)d''a^ 


I  .  2 


dx' 


da, 
-g,{n  -2)^  -q^a,, 


_        d^-'a  d^-'a. 


da. 


—  qn- 


d^  -  ^«-'^'  • 

If  now  we  start  from  the  expression  A^  and  form  successively  the 
expressions 

Ant     A„A„_.^,     A„An-iAn-2y     •  '  •  }     A„A„_.^A„_2  '  •  •  A,, 

we  readily  find,  by  aid  of  the  preceding  equations,  the  following 
values  for /, ,  /, ,  .  .   .  ,  /„  ,  viz.: 

(16)    p,  =  -  {a,  +  a,  -{-  .  .  .  -{-  a„), 

A  =  2a,ay-{n-i)  -j^  -  {n-2)  -^  -  ...  - 


dx 


dx 


dx    ' 


A  =  —  ^  ^i'^j^k— 


\n—i){n—2)d''a,       {n  —  2){n  —  ^)d''a. 


I  .  2 


dx' 


f 


I  .  2 


dx 


-4-.. 


+  («-2)(«,+^3+  .  .  •  +  ^«)  5^  +('^-3X^3+  .  .  •  +^«)^  + 


+  «: 


dx 
dx 


+  ^. 


+ 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.         393 

These  formulae  give  the  coefficients/  in  terms  of  the  coefficients 
^,  and  consequently,  since  the  as  are  functions  of  z^, ,  z', ,  .  .  .  ,  z/„ ,  in 
terms  oi  v^,  v^,  .  .  .  ^  v^.     Write,  for  brevity, 

(17)  Pi  =fi{(i,,  a^,  '  •  '  ,  ««); 

now,  since  p^  ,  p^,   .  .  .  ,  p„  have  the  same  values  whatever  be  the 
chosen  system  of  fundamental  integrals  of  the  differential  equation, 
and  consequently  whatever  be  the  chosen  system  of  the  functions 
V,  it  follows  that  the  functions /"are  invariants. 
\i  a^  ,  a^,   .  .  .  ,  a^  are  constants,  we  have 

(18)  /,  =  —  2^,-,    /,  =  ^Uittj,    A  =  —  '^aiafik,    •  •  • » 

and  therefore /j ,  A»  •  •  •  ^""^  the  coefficients  of  the  algebraic  equa- 
tion 

{t  —  aXt  -  a,)  .  .  ,  {t  —  a„)  =  o, 

having  a^,  a^,  .  .  .  ,  a„  for  roots. 

A  transformation  which  is  frequently  useful  consists  in  changing 
J*  =  o  into  an  equation  whose  integrals  are  those  of  P,  each  multi- 
plied by  the  same  factor,  say  N.  If  2  be  the  dependent  variable  in 
the  new  equation,  then 

(19)  z  =  Nj/        and         ^  =  —z, 
and  the  new  equation  is  therefore 

(20)  ^(^«-)=°- 

Let 

(21)    f,  =  v,,    y,  =  vjv^x,     .  .  .  ,    jj/„  =  vjv^x  .  .  .  fvjx 

denote  a  system  of  fundamental  integrals  of  P  =  o,  and  let 

P  =  A„A„_^  .  .  .  A^ 

■denote   the   corresponding  decomposition   of  P  into  prime  factors. 
In  order  to  multiply  all  these  integrals  by  N  it  is  obviously  only 


394 


LINEAR  DIFFERENTIAL  EQUATIONS. 


necessary  to  change  v^  into  iW, ;  with  this  change  of  v^  the  general 
coefficient  «,•  becomes 

d 


.,.  +  -logAr, 


dy 
dx 


d 

cii  +  -7-  log  A^ 
dx 


J  =  o. 


(22) 

and  A/  =  o  is 

(23) 

We  have  thus  the  identity 

(24)  NP[-^j>j  =  A„'A„'_,  .  .  .A/. 

Suppose  N  =  -—,  then  a/  is 
W 


(25) 

In  particular  let 

(26) 

then 

(27) 

and  we  have 

(28) 

where 
(29) 


a,  -^^\ogV-4-  log  ^• 
'~  dx     ^  dx     ^ 


V=v,,     W=  I, 


^'^°S^="- 


7,p(-^)  =  ^„'^/  , 


^/, 


«,■  +  a, . 


Other  special  cases  are  readily  found. 

We  have,  in  what  precedes,  lowered  the  order  of  the  equation 
P  =  O  by  unity  by  assuming  that  we  knew  a  solution,  j,  =  v^,  oi 
A^  =  o;  that  is,  a  particular  solution  J/^  oi  P  =  o.  By  aid  of  this 
solution  we  find  the  equation 

(30)       A„A„.,  ...A,  =  ^^  +  ^,  -^  +  .  .  .  +  ^„-,^-  =  o 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.         395, 

of  order  «—  i,  and  from  equations  (15)  have  its  coefficients  $',...  qn-^. 
given  as  functions  oi  p^  .  .  .  p„  and  «,  by 

(31)     q,  =  a,-\rp,, 

q-.  =  ^.«i  +  ^^  +  ^^^  ~  ^^  ^  ' 

,     ^     I    (n  —  i)(n  — 2)  d'^a.    .       ,  .  ^«, 


^1 
Since  in  y^j ,  =  -j «i  jFi ,  we  have  from  (i  i) 

(3^)  "■  =  >;<&' 

the  order  of  the  equation  P  =  o  has  been  lowered  by  unity  by  the 
transformation 

whereas  the  ordinary  formula  for  the  lowering  of  the  order  of  the 
equation  is 

(34)  y  =  yjzdx. 

There  is,  however,  a  simple  relation  connecting  these  two  pro- 
cesses.    From  the  equation  y  =  y^f  zdx  we  derive 

(35)  '=iA' 

and  from 

t  ^\  dy        I  dy, 

we  have 

/    \  <^  y 

(■57)  "=^'Z^y.- 


39^  LINEAR  DIFFERENTIAL  EQUATIONS. 

It  follows  now  at  once  that  the  integrals  of  1  * 

11 

(38)  ^„^«-x     ...    ^,  =  0 

obtained  by  one  of  the  two  methods  are  equal  to  those  of  Pi^y^fzdx) 
obtained  by  the  other  when  each  of  the  latter  integrals  are  multi- 
plied by  ji .     If  we  multiply  the  solutions  of 

(39)  P{yjzdx)  =  o 

by  ji ,  which  is  done  by  changing  2  into  —  ,  we  have,  since  the  first 
coefficient  of  Pf^j    /--  c^^'j  is  unity,  the  identity 

(40)  Pf^y.f^dx^  =  A„An-^  .  .  .  A.,. 

If  now  we  divide  the  solutions  of  A„A„_^  .  .  .  A^  =  o  by  y^ , 
which,  by  a  formula  analogous  to  (29),  is  done  by  changing  a^  into 
rtj  —  rtj ,  2  =  2,  3,  .  .  .  ,  ?i,  we  have  a  second  identity, 

(41)  yP{yJj;dx)  =  AJA„'.,  .  .  .  a:, 
where 

(42)  a/  =  tti  —  a, . 

Suppose  the  algebraic  equation  F{^y)  =  o  has  a  root  y  =  y^;  in 

dF 
order  that  r  =  r,  shall  be  a  double  root  we  must  have  -7-  =  o  for 

-^  ay 

y  ^  y^.     A  similar  property  exists  for  the  linear  differential  equation 
„,   ,       d"y  d"-'y 

Suppose  jFi  a  solution  ol  P  =  o  obtained  by  equating  A^  to  zero  in 

P=  A„A„_,  .  .  .  A^=zO. 
In  order  that  y^  shall  be  a  solution  of 

A„A„_^  .  .  .  Aj  =  o 


DECOMPOSITION  INTO    SYMBOLIC  PRIME  FACTORS.         Z97 

where  the  factor  A^  has  been  dropped,  it  is  necessary  and  sufficient 
that  }\  satisfy  the  equation 

*^43)  ''  d^^  +  ^''  ~  ^)-^'  d^^  +  •  •  •  -\-pn-^y  =  O. 

The  proof  of  this  is  very  simple.  The  identity  (40)  shows  that  the 
equation 

^«^H-i  ..--^2  =  0 
is  satisfied  by  j=jy,  if 

(44)  /'(j./^V,r)=o; 
that  is,  if 

(45)  P{xy:)=o. 

This  means  that  /*  =  o  must  have  y  =  xy^  as  an  integral.  Substi- 
tuting this  value  oi  y  in  P  =  o,  and  remembering  that  P{y^  =  o,, 
we  have 

d"-y  d"-y 

(46)  n  -^-;^  +  {n  -    I)A  -^-;^  J^    ,    ,    ,    J^p^_^y^o^ 

which  proves  the  proposition. 

It  is  clear  now  that  if  the  equation 

(47)  A,A»-.  .  .  .  A,  =  o 

has  again  the  integral  y^ ,  it  is  necessary  and  sufficient  that  we  have 

(48)  P  [y,  f  ^-  dx)  =  P{xy:)  =  o  ; 

and  in  general  the  necessary  and  sufficient  condition  that  j^  shall  be 
a  solution  of  P  =  o,  and  of  the  ;z  —  i  equations  derived  from  P  =  o 
by  the  substitution 

dy         I  dy^ 

(49)  -^ -:j-  =^' 

dx       y^dx 


398  LINEAR  DIFFERENTIAL   EQUATIONS. 

is  that  P  =  o  must  admit  of  the  n  solutions 

(50)  y, ,     xy, ,    xy,,  .  .  .  ,x"  y, . 

These  solutions  are  called  conpigate  solutions  of  the  differential 
equation,  and  they  are  analogous  to  the  equal  roots  of  an  algebraic 
equation. 

Suppose  the  differential  equation  P=  o  to  admit  of  k  conjugate 
solutions:  we  can  readily  show  that  P  can  be  decomposed  into  n 
prime  factors  of  which  the  last  k  shall  be  equal.     To  prove  this,  let 

(5 0  y^  =  "^'1 .  J.  =  ^"^^ '  Js  =  ^'^1 ,  .  .  .  ,yk  =  x''-\\ ,  ja.+,  .  .  .y„ 
denote  a  fundamental  system  of  integrals  of  P  =  o,  and  let 

(52)  P=A„An-.   .   •   .   A, 

be  the  corresponding  decomposition  of  P.  If  now  we  calculate  v„, 
t'3 ,  .  .  .  ,  Vi,  by  the  ordinary  formulae,  viz., 

(53)  }\  =  ^^'. '    y.  =  ^^./V-f.     •  •  •  '    yk  =  ^'./^^'.^-*'  •  •  •  f^^'kdx, 
we  find  at  once 

(54)  V,  =  1,     v,  =  2,     .  .  .  ,     Vk=^k—  I, 

and  consequently 

(55)  ^i  =  ^  log  (^V'.  .  •  .  vd  =  ^.  log  iizLlk)  =  ^  log  ^^. , 
or 

(56)  ai  =  a^     for     z  =  2,  3,  .  .  .  ,  k. 

Conversely,  if  P  is  decomposable  into  Ji  prime  factors  of  which  the 
last  k  are  equal,  then  the  equation  P  =^  o  admits  of  k  conjugate 
solutions.     If  the  system  of  integrals 

(57)  J,  =  ^\  ,    y.  =  -^J^'.(^'^',     ■  '  '  ,    yn  =  vjv,dx  .  .  .  fvjx 
corresponds  to  the  decomposition 

A„A„.^  ...  71, 


DECOMPOSITION  INTO    SYMBOLIC  PRIME   FACTORS.         399 

of  P,  we  see  at  once  that  v^,v^,  .  .  .  ,Vk  given  by  equations  (lo) 
(viz., 

^    — -  ^/(«2  —  ax)dx   _    _    _    ^    __  gfiflk  —  ak-i)dx  . 

where  a^  =  a„  =  .  .  .  =:  a^)  are  constants,  and  therefore  jJ'i ,  J2 ,  .  •  •  , 
j'f.  are  of  the  forms 

(58)  Ji  =  ^'>.    7,  =  ^'^,,       -',    yk  =  x''-'v^. 
It  is  easy  to  see  what  takes  place  when  in  the  equation 

A„A„_^  .  .  .  yi,  =  o 
we  have  any  k  consecutive  factors  equal.     Suppose,  for  example, 

(59)  A,+,  =  A,=  .  .  .  =A,; 
the  equation 

(60)  A„A„_,  .  .  .  A,  =  o 

admits  now  the  k  conjugate  solutions 

(61)  v^v^,     xi\v^,     x'v^-o^,     ■  ■  '  ,     x''-'v^v^, 
and  the  proposed  equation  has  the  k  integrals 

(62)  y,  =  vjv^dx,    J3  =  vjxv,dx,     ...     ,  J4+,  =  vJx^'H^dx, 
satisfying  the  relations 

^  -^^  dx  2\         ^       dx  v^  ^  dx    z\  ^ 

that  is,  the  derivatives  of  the  solutions  are  conjugate,  and  not  the 
solutions  themselves. 

We  have  already  remarked  that  the  order  of  the  factors  A  in  the 
decomposition  of 

is  not  arbitrary.     Suppose,  however,  that  for  any  arrangement  of 
these  factors  we  find  the  same  expression  P,  that  is,  we  find  the 


400  LINEAR  DIFFERENTIAL   EQUATIONS. 

same  values  of  the  coefficients  / ;  the  factors  are  then  commutative.. 
We  have  now  to  find  the  conditions  which  must  be  satisfied  in  order 
that  P  can  be  decomposed  into  a  system  of  commutative  prime 
factors. 

To  do  this  compare  first  the  two  differential  expressions 


(64)  A„A„_,  .  .  .  A,A, 
and 

(65)  A„.,A,,  .  .  .  A,A,. 

Call  for  brevity  the  common  part  of  these  expressions  A,  that  is, 

(66)  A„_2A„_^  .  .  .  A^A^  =  A, 
and  we  have  at  once 

d^A  dA       I  dn      \ 

(67)         A  An- A    =  ^  -  (^„_,  +  ^„)   -  +  ^^„«„_,   -   -^j  A, 

......      d'A       ,         ^         dA       I  da„\  ^ 

(68)     A„_,AnA  =  -^  —  {a„_,  +  «„)  ^  +  \a„a„_,  —  -^:)A. 

In  order  that  these  may  be  identical,  or  that  (64)  and  (65)  may 
give  the  same  set  of  values  of  the  coefficients/,  we  have 

da„        da„_, 

that  is,  a„  —  a„_.,  =  constant.  We  have  then  that,  in  order  that  in 
a  composite  differential  expression  we  may  be  able  to  change  the 
order  of  the  first  two  prime  factors,  the  difference  of  the  coefficients 
a  of  these  factors  must  be  constant.     Again,  compare 

(70)  A„An-iA„-2A„.An-4    •    •    '    A^ 

and 

(71)  A,An-iA„_^A„_2An-A    '    •    •    A^^ 


.DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.         4OI 
By  aid  of  the  preceding  theorem  it  is  clear  that  if 

(72)  A„_2A„_^A„_^  .  .  .  A^,  =  S, 

and 

{73)  ■^K-3-^«-2'^»2-4      •       •       •      A^    ,      =      T, 

are  identical,  we  must  have 

(74)  ««-2  —  ««-3  =  const. 

In  this  case  (70)  and  (71)  are  obviously  also  identical.  Conversely, 
if  (70)  and  (71)  are  identical,  then  (72)  and  (73)  are  also  identical. 
In  fact,  we  have,  if  (70)  and  (71)  are  identical, 

(75)  A„A„.,S-A„A„_,T=A„A„_,{S-  T) 
identically  zero ;  but 

(76)  A„A^_^{S-  T)  =  ^'^^~  ^^  -  K..  +  ««)^^ 


Suppose  this  is  not  identically  zero,  that  is,  suppose  5  and  T  are 

dh' 
not  identical,  and  let  R  -^-^  be  the  first  term  \nS —  7" which  does  not 

d''+y 
vanish  ;  then  in  {76)  we  shall  have  the  term  R  -tt+I  '  which  will  not 

cancel  with  any  other  term,  which  is  contrary  to  hypothesis.  Hence 
6"  and  T  are  identical.  We  see  at  once  now  that  if  in  a  differential 
expression  we  can  change  the  order  of  any  two  consecutive  factors, 
we  must  have  that  the  coefficients  of  these  factors  differ  only  by  a 
constant.     We  are  thus  led  to  the  general  result : 

In  order  that  the  factors  of  a  differential  expression  wJiicJi  is  com- 
posed of  prime  symbolic  factors  may  be  commutative,  it  is  necessary  and 
sufficient  that  the  differences  of  the  coefficients  of  the  factors  taken  in 
pairs  shall  be  constants. 


402  LINEAR  DIFFERENTIAL   EQUATIONS. 

Consider  now  the  two  equations  A/,  =  o  and  Ai  =  o;  that  is, 

dy  dy 

{77)  ^-a,y  =  o,     --a,y  =  o', 

of  which  j/k  and  yi  are  the  general  integrals.  In  the  first  of  these 
write  y  =.  y^  and  we  have 

(78)  ^  ~  (""^  ~  ""^^  =  °  ' 

if  now  ttk  —  «i  is  a  constant,  we  have  for  z  the  value 

(79)  -s^  =  ^^""> 

where  C  and  a  are  constants  ;  conversely,  if  3  is  of  this  form,  a^  —  ai 
is  a  constant.  The  condition,  therefore,  that  the  difference  a,,  —  ai 
shall  be  a  constant  gives  for  the  ratio  of  the  integrals  of  equations 
ijf)  the  value 

(80)  -  =  Ce--. 
^     ^          .  yi 

From  this  results  the  following  theorem  : 

If  the  factors  of  a  differential  expression  composed  of  prime  symbolic 
factors  are  commutative,  it  is  necessary  and  sufficient  that  the  ratios  of 
pairs  of  integrals  of  corresponding  factors  shall  be  of  the  for7n  Ce°-^ , 
where  C  and  a  are  constants. 

The  special  forms  of  this  result  for  equations  with  constant 
coefficients  are  readily  found ;  this  will,  however,  be  left  as  an  ex- 
ercise. 

It  is  evident  that  a  differential  expression  composed  of  prime 
commutative  factors  vanishes  if  an  expression  composed  of  one  or 
several  of  its  factors  vanishes.  By  aid  of  this  remark  we  will  proceed 
to  determine  the  form  of  the  differential  expression 

when  it  is  decomposable  into  commutative  prime  factors.  We 
remark  first  that  the  coefficients  p  are  symmetric  fuYictions  of  the 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.        403 

coefficients  a  of   the  commutative  factors.     These  coefficients  are 
found  from  equations  (16)  when  we  write  in  the  latter 

^     ^  dx  ~  dx         *  *  "         dx  '' 

and  so  for  the  higher  derivatives. 

Suppose   now  P  to  be  decomposable  into  commutative  prime 
factors  ;  then 

(82)  P  =  A,An-.  .  .  •  A,, 

where  «,•  —  ^k  =  const,  for  t,  k  =  i,  2,  .  .  .  ,  n. 

Consider  a  solution  j^  of  A^  =  o;  j/k  is  also  a  solution  of  P=o. 
Now  we  know  that 

(83)  --p{m)  =  a\^a'„_,  .  .  .  a/ 

where 

(84)  a/  =  tti  —  ak,         i=  1,2,  .  .  .  ,  n. 

It  follows  then  that  the  expression 
has  its  coefficients  constants,  and  so 

wl-wssre  g,,  .  .  .  ,  g„_^  are  constants.     Changing  1/  into  --  ,  we  have 

yk 

m  P{y)==y.Q{j^. 

Therefore  :  Every  differential  expression  P{y)  which  is  decomposable 
into  commutative  prime  factors  is  of  the  form 

<^7)  P{y)=y.Q{{-); 


404  LINEAR  DIFFERENTIAL   EQUATIONS. 

where  yk  is  a  function  of  x  a?id  Q{y)  is  a  linear  differential  expressiofv. 
with  constant  coefficients. 

Conversely :  Every  linear  differential  expression  of  the  form 

y 


PW=..G(^;) 


is  decomposable  into  commutative  prime  factors. 
We  have  in  fact 

(88)  G(J)  =  ^«^«-:    '    '   •   B,, 

where  the  coefficients  b  are  constants ;  now  we  know  that 

(89)  J*S  (v)  =  ^«'^»'-x    •    •    •    ^/> 

where  the  coefficients  b'  have  the  form 


(90) 


and  so  the  differences  bl  —  bj,  i,j  ^=  i,  2,  .  .  .  ,  n  are  constants^ 
and  consequently  the  factors  B'  are  commutative.     It  is  therefore 

necessary  and  sufficient  that  P{y)  should  have  the  form  y,,Q  [—-]   in 

order  that  it  may  be  decomposed  into  commutative  prime  factors. 
It  is  now  easy  to  see  what  in  the  supposed  case  is  the  form  of  the 
integrals  of  P  =:  o.  Let  w^,  w.^,  .  .  .  ,  w„  denote  a  system  of  in- 
tegrals of  ^  =  o ;  one  of  these  is  of  course  a  constant,  since  Q  con- 
tains no  term  in  y,  and  the  others  are  of  the  form  ;ir V^-^.  The  equa- 
tion Q{  —  j=o,or  P{y)  =  o,  will  then  have 

as  a  system  of  linearly  independent  integrals.  Finally,  then,  if  P  is 
decomposable  into  a  system  of  commutative  prime  factors,  and  if 
yk  is  a  function  which  causes  one  of  these  factors  to  vanish,  P  =  o 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.        405 

has  a  system  of  linearly  independent  integrals  of  the  form  jprV^-^ , 
the  integral  y^  being  included  in  this  system.  As  an  example,  let  us 
find  the  condition  to  be  satisfied  in  order  that  the  differential  ex- 
pression 

{9.)  ^^)  =  ^+a|,+a^ 

of  the  second  order  may  be  decomposed  into  commutative  factors, 
say 

(92)  P=A,A,  =  A,A.- 
We  have  directly 

(93)  A  A.  =  g  -  {a.  +  -.)%  +  [a^a,  -  ^')  J, 
giving 

(94)  P.=  -  {a,  +  a,),    /,  =  a,a^  —  ^, 
with  the  condition 

dx       dx 


(95) 
Now 


da,      da^      \    d  i  dp, 

<96)  ^  =  ^  =  2^(^'  +  ^^)  =  -2^' 

and 

<97)  ^'^^=^^-2^  = 

consequently 

^9^^  ""l^--^=  +  2^  +  T'' 

but  ^1  —  «2  =  const.,  therefore 

(99)  ^^-2^-7-^'°"''" 


406  LINEAR  DIFFERENTIAL  EQUATIONS. 

which  is  the  required  necessary  and  sufficient  condition.  If  the  two* 
commutative  factors  are  equal,  then 

I  dp,      p^ 
(lOo)  ^^-2Z^-T^°- 

If  we  transform  the  equation 

d'^y   ,       dy    ^ 
(loi)  ^^+^"^+^^-^  =  ° 

by  removing  in  the  usual  way  its  second  term,  we  have 

(I02)  _-  +  /,:=o, 

where 

then  if  the  seminvariant  /  is  a  constant  we  can  always  decompose 
d^y   ,       dy    , 

into  commutative  prime  factors. 

We  will  proceed  now  to  apply  the  results  arrived  at  concerning- 
decomposition  into  prime  factors  to  the  subject  of  the  regular  in- 
tegrals of  a  linear  differential  equation.  We  will  recall  first  the 
properties  of  the  integrals  of  the  equation  of  the  first  order, 

(104)  -^  -  ^j  =  o, 

where  «  is  a  uniform  function  of  x  containing  both  negative  and 
positive  powers  of  x^  and  where  the  double  series  so  formed  is  con- 
vergent in  the  region  of  x  =  o.  In  order  that  this  equation  shall 
have  only  regular  integrals  it   is  necessary  and  sufficient  that  a  shall 

be  of  the  form  —  ,  where  tv  is  a  uniform  function  of  x  containing  only 

X 

positive   powers  of  x,  and  which   may  contain   a  power  of  ;ir  as  a 


DECOMPOSITION  INTO   SYMBOLIC  PRIME   FACTORS.        407 

factor,  that  is,  may  vanish  for  x  ^  o.  If  these  conditions  are  satis- 
fied, the  integrals  of  (104)  are  of  the  form  y  =  xPip{x),  where  p  is  the 
single  root  of  the  indicial  equation,  and  ip{x)  is  a  uniform  function 
of  X  which  does  not  vanish  for  x  =  o.  If  «  does  not  contain  only- 
positive  powers  of  x,  and  if  a  becomes  infinite  of  order  v  -\-  1  for 
X  ^  o,  the  integrals  are  of  the  form 

■   ^  +  ^,+  .  .  .  +- 

^  __  ^^        X-  ^^X''lp{x), 

p  and  rp{x)  being  characterized  as  before. 

When  a  contains  only  positive  powers  of  x  we  will  call 

dy       a 
dx      X 

a  regular  prime  factor  of  a  differential  expression.  It  is  easy  to  see 
from  equations  (16)  that  a  differential  expression  P  which  is  made  up 
of  n  regular  prime  factors  is  of  the  form 

(,05)  d^      PJ^d-^ 

where  PS^),  .  .  •  ,  Pn{x)  are  holomorphic  functions  of  x  in  the 
region  of  ;ir  =  o,  and  may  vanish  for  ;ir  =  o ;  and  further,  that  such 
an  expression  is  put  into  its  normal  form  when  we  multiply  it  by 
x'^ .  We  have  shown  that  if  a  given  differential  expression  is  com- 
posed of  differential  expressions  which  are  in  the  normal  form,  the 
given  differential  expression  will  itself  be  in  the  normal  form,  and  its 
indicial  function  will  be  the  product  of  the  indicial  functions  of  its 
components.  Bearing  this  result  in  mind,  we  will  proceed  to  the 
consideration  of  expressions  of  the  form 

d»y    ,        d''-'y    , 

which  is  not  in  the  normal  form,  but  where  the  coefificients  p  are 
developable  in  series  going  according  to  integral  powers  of  x  and 
containing  only  a  finite  number  of  negative  powers  of  x.     Suppose, 


4o8 


LINEAR  DIFFERENTIAL  EQUATIONS. 


first,  that  a  differential  expression  P  is  composed  uniquely  of  regular 

factors,  say 


(io6) 

where  the  factors 


P  —  A„A„.j  .  .  .  A^, 


(107) 


Ai  = y    (?  =  I,  2,  .  .  .  ,  w) 

ax        X 


are  all  regular,  and  let 

y.  —  "^M    y.  =  "vj^'^/ix,     .  .  .  ,    X,  =  'vjv^x  .  .  ,  fvjx 

denote  the  fundamental  system  of  integrals  corresponding  to  this 
decomposition  of  P.  How  shall  the  factors  A  of  (106)  be  modified 
in  order  that  their  new  expressions  shall  be  in  the  normal  form,  and 
that  at  the  same  time  they  will  give  P  in  its  normal  form,  viz.  x^P  ? 
This  question  is  readily  answered.  Multiply  each  factor  A  by  x, 
and  let  A"  denote  the  product ;  A"  is  of  course  in  the  normal  form, 
and  since  the  equations 

(108)  a;'  =  o,    A/'=  v,v,,    A," A,"  =  v,v,v,,     .  .  . 

admit  the  integrals 


(109) 


^»'      '^'f'x^^'      '^'f  'x^^f'x^^'     '  •  • 


we  see  that 

(no)  P"  =  AJ'AJ'.,  .  .  .  A:'  =  o 

has  the  same  integrals  for  a  fundamental  system.     Suppose  then  we 

add  to  the  coefficients  — -  the  quantities 


(III) 


dx 


logf  ;f ' 


t  —  \ 


or,  what  is  the  same  thing,  increase  the  functions  ai  by  the  numbers 
i  —  i;  then,  as  already  shown,  the  functions  v^,  v^,  .  .  .  ,  v„  will  be 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.         409 

•each  multiplied  by  x\  the  factors  A"  are  now  changed  into  factors 
A'  of  the  form 

<II2)  A'  =  ^^-(^.  +  ^-l)j, 

and  these,  on  being  combined  in  the  usual  way,  give  rise  to  the  ex- 
pression 

<ii3)  P'=A:a:_^.  .  .  A,', 

which  equated  to  zero  has  for  a  fundamental  system  of  integrals 

JJ^i  =  "^i^    J2  =  '^'./^.^-^^     ■      •   ,    yn  =  vjv.,dx  .  .  .fvjx; 

but  these  constitute  a  fundamental  system  of  integrals  of  P  =  o,  and 
consequently,  since  the  first  coefficient  of  P  is  unity  and  that  of  P' 
is  ;ir",  we  have 

(114)  P  =  x-^P'. 

We  have  thus  the  theorem  :  Having  given  a  differeiitial  expres- 
sion composed  uniqjiely  of  regular  factors  of  the  form 

dy  Ui 

dx  X 

it  can  be  at  once  throivn  into  the  normal  form  by  adding  the  numbers 
i  —  I  to  each  of  the  quantities  ai ,  and  tJien  putting  each  of  the  regular 
factors  into  the  normal  for 7n  by  multiplying  them  by  x. 

Consider  now  the  composite  expression 

(115)  P=QD, 

where  Q  and  D  are  linear  homogeneous  differential  quantics  of 
orders  n  —  s  and  s  respectively,  each  of  which  has  unity  for  its 
first  coefficient  and  rational  functions  of  x  for  the  remaining-  coef- 
ficients.  P  is  then  of  the  same  form  as  its  components  Q  and  D, 
.and  is  of  order  71.  Let  x  ^,  x^,  x''  denote  the  powers  of  x  by  which 
we  must  multiply  the  differential  expressions  P,  Q,  D,  respectively, 
in   order  to  put  them  into  their  normal  forms.     We  will  now  find 


4IO  LINEAR  DIFFERENTIAL  EQUATIONS. 

the  relation  connecting  the  indicial  functions  g{p),  k{p),  h{p)  of 
P,  Q,  D.     Make  in  D  the  substitution 

y  =  x^'z  ; 

the  resultant  expression  D'  has  the  normal  form,  a  fact  which  is 
seen  at  once  when  we  recall  the  law  of  formation  of  the  coefficients 
in  D' .  The  expression  Pi^)')  now  becomes  QD' ,  or  P{x^2).  Now 
put  Q  into  its  normal  form  Q' ,  by  multiplying  it  by  x'',  and  we  have     | 

(1 1 6)  x''P{x^y)  =  Q'D'. 

The  expression  x''P{x^f)  is  now  in  the  normal  form,  and,  denoting 
hy  g'{ij),  k'{p),  h'{p)  the  indicial  functions  of  x''P{x''y),  Q' ,  D'  respec- 
tively, we  have 

(117)  g'{p)  =  k\iS)h\p). 

We  have  of  course  the  relation  g  =.  k  A;-  Ji  among  these  exponents. 
We  have  already  shown  that  the  indicial  functions  of 

A'*P(;r'»,     or     P{x''y),     and     D{x^')^ 

are  immediately  deducible  from  those  of  P(jj')  and  D{y)  by  changing 
p  into /3 -|- // ;  consequently 

(118)  g\p)  =  g{p  +  //),      /i\p)  =  h{p  +  h\ 

and,  since  k\p)  =  /'(p), 

(119)  g{p^  h)  =  k  {p)h{p  +  h). 

Changing  now  p  into  p  —  //,  we  have  the  identity 

(120)  g{p)  =  h  {p)k{p  -  h). 
In  particular,  if  D  is  of  the  form 

(121)  • — --  -4-  — —  — --  +  .  .  .  4-  — —  y, 
we  have 

(122)  g{p)  =  Kp)Kp  —  ^\ 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.         4II 

since  h  is  now  equal  to  s.  By  proceeding  to  the  case  where  P  is  of 
the  form  P  =  QDE,  then  to  the  case  where  P  =  QDEF,  etc.,  we 
will  easily  arrive  at  the  following  general  theorem :  If  P  denote  a 
differential  expression  of  order  n  composed  of  6  differential  expres- 
sions, viz., 

P  =  DeDe-,  .  .  .  D,, 

where  the  components  D  are  linear  homogeneons  differential  expres- 
\  sions  whose  first  coefficients  are  each  iinity  and  whose  remaining  coef- 
ficients are  rational  functions  of  x,  and  if  x''i  is  the  power  of  x  by 
wJiicJi  it  is  necessary  to  multiply  Di  in  order  to  reduce  it  to  the  normal 
form  :  then,  denoting  by  g  (p)  the  indicial  function  of  P,  and  by  hi{p) 
J  he  indicial  function  of  Di,  we  have  the  identity 

(123)  g{p)  =  K{p)K{p  -  Jiyh{p-K  -  K)  •  .  . 

h„{p  —  h,  —  h^—  ,  .  .  —  h,),. 

and  also 

.      K  +  K-\-  '  '  '  +  K  =  g, 

where  g  is  the  exponent  of  the  factor  x^^,  by  zvhicJi  zve  have  to  mtiltiply 
P  in  order  that  the  product  may  have  the  normal  form. 
If  all  the  D's  have  all  their  factors  regular,  we  have 

(124)  h,=  h,=  .  .  .   =  h,  =  I, 
and  consequently 

(125)  g{p)  =  Hp)h,{p  -  i)  .  .  .  h„{p  -  n-\-  i). 

We  deduce  at  once  from  this  general  theorem  the  corollary:  The 
degree  of  the  indicial  function  of  P  is  equal  to  the  sum  of  the  degrees 
of  the  component's  functions  D. 

We  will  now  take  up  the  general  question  of  the  decomposition 
of  the  differential  expression 

d"y    ,        d^'-'y 

where  the  coefficients/  are  developable  in  double  series  proceeding 
according  to   positive   and  negative  integer  powers  of  x.     The  re- 


412 


LINEAR   DIFFERENTIAL  EQUATIONS. 


striction  to  a  finite  number  of  negative  powers  of  x  will  not  be  here 

applied. 

We  know   that  P  =:  o  always  admits  an   integral   of    the    form 

y^  =  x''(p{x),  where  (p{x)  is  a  double  series,  similar  therefore  to  the 

log  s 
coefficients  p  and  r  = .,  where  5  is  a  root  of  the  characteristic 

equation  for  the  point  x  =  o. 

Suppose  now  P  =  o  to  admit  an  integral  of  the  form  j,  =  xPip{x) 
(that  is,  a  regular  integral),  where  ip{x)  is  a  holomorphic  function  of 
x  and  ^(o)  is  not  zero.     Make  the  substitution 


and  we  have 
(126) 

where 


y  -  yjsdx, 

+   •  •   •   +  ^«-i^  =  o, 


^1  = 


J. 


'(127)    \ 


^.  =  — 


';/(;/  —  i)  dy^ 


dy. 


J.L     1.2 


^.^+^^^-^>'^'+^'^'. 


The  forms  of  the  remaining  coefificients  are  easily  found.  It  is  easy 
to  see  that  these  coefficients  q  are,  like  the  coefificients/,  uniform  in 
the  region  of  ;tr  =  o.  In  fact  the  coefificients  q  are  made  up  from 
the  p's  and  sums  of  the  form 


0=  A 

^  c^x 


Mh-a) 
—a  ^  W  


where  a  is  an  integer.  Each  of  these  sums  is  then  uniform  and 
continuous  in  the  region  oi  x  =  o,  but  of  course  having  ;ir  =  o  as  a 
critical  point.  The  same  conclusion  then  holds  for  the  coefificients  q. 
It  is  important  to  remark  here  that  though  this  result  has  been 
arrived  at  by  assuming  7p(x)  to  contain  only  positive  powers  of  x, 
this  restriction  is  not  necessary:  it  suffices  that  tp{x)  be  developable 
in  a  double  series  proceeding  according  to  positive  and  negative 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.        413 

powers  of  x  and  convergent  in  the  region  of  ;f  =  o,  and  further  that 
'^{x)  shall  have  no  zeros  infinitely  near  ;i;  =  o.  The  necessity  for 
this  last  remark  is  easily  shown  as  follows :  Suppose  fix)  to  be  a 
uniform  function  of  x  in  the  region  of  x  ^  o,  and  suppose  the  point 
;tr  =  o  to  be  an  isolated  essential  singular  point  for  f{x)\  i.e.,  there 
must  exist  no  pole  or  essential  singular  point  infinitely  near  the 
point  X  =■  o.  In  this  case  the  function  fix)  is  developable  in  a 
double  series  proceeding  according  to  positive  and  negative  powers 
of  X  and  convergent  in  the  region  of  -r  =  o.  The  derivatives 
f',f",  .  . .  are  of  course  developable  in  the  same  way.  We  require 
for  our  purpose,  however,  that  this  shall  also  be  true  for  the  loga- 

rithmic  derivative  -yr  and  for  the  functions  —7-.    l(f{x)  has  no  zeros 

f'Yx) 
infinitely  near  x  =:  o,  then    ,..  .   can  have  no  point  of  discontinuity 

infinitely  near  x  =  o,  and  is  consequently  developable  in  a  conver- 
gent double  series  in  this  region.  If,  however,  y(;r)  =  o  has  an  in- 
finite number  of  solutions  in  any  region  however  small  of  ;c  =  o,. 


then  -?7-T  will  have  in  this  region  an  infinite  number  of  poles,  and 
f{x)  ^  ^ 

consequently  will  not  be  developable  in  the  same  way  asf{x).   This 

shows  us  at  once  the  necessity  for  the  above-mentioned  restrictioa 

upon  the  function  ip{x).     Suppose  now 

(128)  P=A„A„_,.  .  .  A,, 
and  let 

(129)  y,=  V,,  y,  =  vjv^dx,  .  .  . ,  jj/„  =  I'Jv^x  .  .  .  fv,4x 

denote  the  system  of  fundamental  integrals  corresponding  to 
this  decomposition  of  P.  As  the  auxiliary  equations  of  which 
t\,%\y  .  .  . ,  z'„  are  integrals  are  of  the  same  form  as  Py  we  can,  and 
will,  choose  v^,  v^,  .  .  .  ,  v^  oi  the  forms 

(130)  V,  =  x'-i  <f),{x),     v^  =  x"-^  (plx),     .  .  .  ,     v„=  x''ncf)„{x). 
We  have  now 

(i  3  0     «»•  =  ^  log  {v,  V,.  .  .   v^  =  -^  log  {x-^+-^+  •  •  •  +  ^-^-0,0, .  .  .  0,). 


414  LINEAR  DIFFERENTIAL  EQUATIONS. 

If  now  the  functions  0  have  no  zeros  infinitely  near  the  point 
X  =  O,  it  follows  that  the  coefificients  a  are,  in  the  region  of  x  ■=  o, 
continuous  and  monogenic  functions  having  this  point  as  a  critical 
point ;  they  are  also  uniform  :  for,  when  the  variable  turns  round  the 
point  X  ^=  O,  the  quantity  under  the  logarithmic  sign  is  multiplied 
]-,y  ^277/(^1+^2+. .  .+r«)^  and  therefore  its  logarithmic  derivative  is  un- 
altered. The  coefificients  a  have  therefore  the  same  properties  as 
the  coefficients/.  We  have  thus  the  result:  If  the  functions  (jiix) 
•admit  of  no  zeros  infinitely  near  the  point  x  =  o,  it  is  possible  to  decom- 
pose the  differential  expression  P  into  prime  factors  having  uniform 
coefficients  of  the  form 


2  CiX\ 


where  i  is  an  intes'er. 


Resume  now  the  particular  case  where  the  coefficients/'  contain 
only  a  finite  number  of  negative  powers  of  x.  The  decomposition 
of  P  into  prime  factors  of  the  form 

dy  +„~ 

y  2  dXi 

dx  _  CO 

leads  to  two  important  propositions  : 

(A)  If  P  is  decomposable  into  prime  factors  of  the  form 

—  y   '2  dx", 


dy  + 


dx 

the  degree  y  of  its  indicial  function  is  equal  to  the  total  number  f  of 
regular  factors  ivJiich  enter  into  this  decomposition. 
To  prove  this,  we  will  consider  the  decomposition 

P=A„A„_,.  .  .  A,, 

which  contains  /  regular  factors.  Let  us  neglect  for  a  moment,  in 
factors  which  contain  an  infinite  number  of  negative  powers  of  x,  all 
of  the  powers  of  x-'  which,  in  absolute  value,  are  greater  than  a 
given  arbitrary  number  k,  and  let 

P\  =  B,A.-.-  .  .  ^w 


DECOMPOSITION  INTO    SYMBOLIC  PRIME  FACTORS.         415 

•denote  the  modified  form  of  P.  The  expression  P'  is  of  the  same 
character  as  P,  and  the  regular  factors  in  P'  of  course  coincide  with 
they  regular  factors  in  P.  If  now  y'  denote  the  degree  of  the  in- 
dicial  function  of  P' ,  we  know  that  y'  is  equal  to  the  sum  of  the 
degrees  of  the  indicial  functions  of  the  factors  B.  Now  the  ir- 
regular factors  B  have  constants  as  their  indicial  functions ;  the 
regular  factors  B  are  all  of  their  first  order,  and  so  their  indicial 
functions  are  of  the  first  degree,  consequently  we  have  y'  =y.  This 
equality  evidently  exists  however  great  k  may  be  ;  we  may  then 
increase  k  indefinitely,  and  since  y'  has  y  for  its  limit  we  have, 
finally,  y=^j.  The  proposition  A  is  thus  proved.  As  a  corollary  it 
may  be  remarked  that  the  number  of  regular  factors  which  enter 
into  a  decomposition  of  P  into  prime  factors  of  the  form 

ax  _  CO 

is  constant  whatever  be  the  chosen  method  of  decomposition.  In  a 
decomposition  P=.  A„A,i_j  .  .  .  ^j  let  cr  denote  the  greatest  possible 
number  of  consecutive  regular  factors  which  such  a  decomposition 
can  have  when  we  count  back  from  the  factor  A^ ;  that  is, 

/J         /]  A 

■^J- a-  J       ■'^iT  — I  >        •    •    •   ■)       -^^  I 

are  all  the  possible  consecutive  regular  factors  which  can  appear  at 
the  right-hand  end  of  the  decomposition  A,^A„_J^  .  .  .  A^ .  Our  sec- 
ond proposition  is  now : 

(B)  T/ie  number,  s,  of  linearly  independent  regular  integrals  of  the 
£quation  P-  =  o  is  equal  to  the  greatest  number,  a,  of  consecutive  reg- 
ular factors  which  can  terminate  a  decomposition  of  P  into  symbolic 
factors  of  the  form 

dx  _  00 

This  is  readily  proved.  Since  P=  o  has  s  regular  integrals,  it  has  at 
least  one  integral  of  the  form  j,  =  v^  =  xp^xp^{x),  where  rf\{x)  is  in 
the  region  of  ;ir  =  o  a  holomorphic  function  of  x,  and  ^'(o)  is  not 
equal  to  zero.  Make  now  the  substitution  j/  =  v^fzdx,  and  we  have 
a  differential  equation  Q{2)  =  o  of  order  n  —  i  having  .y  —  i  regular 
integrals.     Q{2)  =  o  has  then  at  least  one  integral  of  the  form 

2,  =  V,  =  xp^iplx). 


41 6  LINEAR  DIFFERENTIAL  EQUATIONS. 

Make  now  a  substitution  similar  to  the  prececTing  one,  and  continue 
this  process  until  we  arrive  at  an  equation,  say  H{ii)  =  o,  which  has 
one  regular  integral  of  the  form  v^  =  x^^^iplx).  Now  make  the  sub- 
stitution u  =  v^f  vds ;  the  reduced  equation  will  have  at  least  one 
integral  of  the  form  x''(p{x)  which  contains  no  logarithms.  By  an 
analogous  series  of  reductions  we  will  obtain  the  functions 

o 

and  can  then  give  the  decomposition  of  P  into  prime  factors  corre-     | 
spending  to  the  system  of  fundamental  integrals 


/i  =  ^^.    J=  =  '^\f^^',dx,     .  .  .  ,     >'„  =  vjv^dx  .  .  .  fvjx. 

Let  Jl    =  yi„A„_j  .  .  .  A^ 

denote  this  decomposition,  where 

dv  +  " 

A  =-j--y2C,x\ 
dx  _  00 

and  where  the  equations 

A,  =  o,    A,=  o,    .  .  .  ,    As  =  o 

admit  the  regular  integrals 

v^  =  xp^tp,  ,     v,v,  —  xP^+Pvp4^^,     .  .  .  , 

?\v^  .  .  .  V,  =  ,rPi+P»+-  •  ■+P'4-\4\  •  •  '  fs' 

Since  these  last  s  factors  are  regular,  we  have  cr  ^  s.  It  is  easy  to 
see,  however,  that  we  will  not  have  o"  >  ^.  Suppose  in  fact  that  it 
is  possible  to  so  decompose  P  into  prime  factors  of  the  iorm 

dx-^^f'"' 

that  the  last  s -\- t  factors  shall  be  regular,— say 

P  =  £„B„_,  .  .  .  B,+,  .  .  .  B,B, . 
Denote  by 


DECOMPOSITION  INTO    SYMBOLIC  PRIME  FACTORS.         417 
the  respective  solutions  of  the  equations 

B,=  o,    B,  =  o,    B,  =  o,     .  .  .  ,    B,+t  =  o. 

These  solutions  are  regular  and  of  the  form  xPip[x),  where  ip{x)  is  a 
holomorphic  function  of  x  in  the  region  of  ;r  =  o  and  ^(o)  does  not 
vanish.     It  follows  then  that  the  ratios, 


w^w^  .  .  .  w,-_i 
that  is,  the  integrals, 

of  the  auxiliary  equations,  deduced  the  one  from  the  other  by  the 
ordinary  substitutions,  are  regular  integrals  of  the  form  x^ip[x). 
Now  since  the  equation  which  gives  iv.j^i  has  at  least  one  regular 
integral,  that  which  gives  Ws+<_i  must  have  at  least  two  and  that 
which  gives  'w,j^t-2  must  have  at  least  three  regular  integrals,  etc., 
finally,  then,  the  equation  which  gives  w^ ,  that  is,  P=  o,  must  have 
at  least  s -{-  t  regular  integrals  ;  therefore,  since  this  is  contrary  to 
our  hypothesis,  we  must  have  exactly  (X  =  j.  It  is  easy  to  see  that 
proposition  (B)  is  still  true  when  the  prime  factors  are  not  restricted 
to  be  of  the  form 

ax         _  00 

From  (A)  and  (B)  we  derive  at  once  the  following  theorems : 

{a)  If  P  =  o  has  all  of  its  integrals  regular,  P  is  decomposable 
into  n  regular  prime  factors. 

{b)  If  P  is  decomposable  into  n  regular  factors,  /•  =  o  lias  all  of 
its  integrals  regular. 

(f)  IfP^o  has  all  of  its  integrals  regular,  the  degree  of  its  in- 
dicial  equation  is  equal  to  its  order. 

{d)  If  the  degree  of  the  indicial  equation  of  P  is  equal  to  the  order 
of  P,  then  the  equation  P  =  o  has  all  of  its  integrals  regular. 

The  following  theorem  is  easily  established  : 


41 8  LINEAR  DIFFERENTIAL   EQUATIONS. 

{e)  //"/'  =  o  has  all  of  its  integrals  regular,  it  has  a  fundamental 
system  of  integrals  belonging  to  exponents  zvJiich  are  roots  of  its 
indicial  equation. 

U  P  =  o  has  all  of  its  integrals  regular,  P  is  decomposable  into 
n  regular  factors,  viz., 

P  =   -^n-n-n-i     •    •    •    -"i  > 

and  from  (125)  we  have  the  relation 

g{p)  =  fh{p)K{p  -  i)  •  .  .  hj^p  -  ;«+  i), 

connecting  the  indicial  functions^  and  Ji  of  P  and  its  factors  A.  If 
now  we  write 

^,  =  o,     ^,  =  o,     .  .  .  ,     A„  =  o, 
we  see  that  the  integrals 

of  these  equations  belong  respectively  to  the  exponents 

where 

P\  1        Pi  '        '     •     '     ■>        Ph 

are  the  roots  of  the  indicial  equation  g{p)  =  o.  It  is  obvious  then 
that  the  integrals 

y^  =  "^'^  .     7^  =i\fv^dx,     .  .  .  ,    y^=  vjv^x  .  .  ./z>„dx 

o{  P  =z  o  belong  to  the  exponents  p^ ,  p^ ,  .  .  .  ,  p„  respectively.  In 
the  case  where  the  integrals  of  the  difTerential  equation  are  not  all 
regular,  propositions  (A)  and  (B)  give  the  following  theorem : 

T/ie  number  of  linearly  independent  integrals  ofP  =  o  is  at  most 
equal  to  the  degree  of  its  indicial  equation. 

The  form  which  the  expression  P  must  have  in  order  that  the 
equation  P  =  o  shall  have  s  regular  integrals  is  given  in  the  follow- 
ing theorem : 

Li  order  that  P  =  o  shall  have  s  linearly  independent  regular  inte- 
grals it  is  necessary  and  sufficient  that  P  can  be  put  in  the  form 

P  =  QD, 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.        419 

■ivhcre  Q  and  D  are  respectively  of  the  orders  n  —  s  and  s,  and  are  ex- 
pressions of  the  same  kind  as  P  {that  is,  the  first  coefficient  in  each  is 
unity  and  the  remaining  coefficients  are  rational  functions  of  x),  and 
zuhere  Z>  =  o  lias  all  of  its  integrals  regular  while  Q  =.  o  has  no  sucJi 
integral. 

We  will  show  first  that  this  condition  is  necessary.  We  have 
shown  that  if  /*=  o  has  j  regular  integrals,  it  admits  of  a  decom- 
position 

P  =   AnAn-1    .    .    .    As-\-iAs     .    .    .    A^ 

into  prime  factors  of  the  form 


dy 


+00 


ax       •'  _„ 
where  the  last  s  factors  are  regular.     Write  now 

As  .  .  .  A^A^  =  D,     A,iA,i_i  .  .  .  As+i  =  Q', 
then 

P  =  QD. 

Since  D  is  of  order  s  and  contains  only  regular  factors,  we  have 

dx"    ^     X      dx'-^^  '  '  ''^    X'    -^^ 

the  form  of  the  P's  being  known,  and  B  =  o  has  then  all  of  its 
integrals  regular.  The  expression  Q  is  of  order  n  —  s.  Effecting 
now  the  operation  QD  and  identifying  the  result  with  P,  we  obtain 
a  system  of  equations  which  show  (under  the  above-mentioned 
restriction  as  to  the  functions  cp)  that  the  coefficients  of  Q  are,  like 
those  of  P  and  D,  rational  functions.  Further,  Q  has  no  regular 
integral ;  for,  if  it  had,  we  could  terminate  a  decomposition  of  Q  by 
at  least  one  regular  factor,  and  so  P,  =  QD,  would  have  more  than  s 
regular  integrals,  which  is  impossible  by  proposition  (B).  Second, 
this  condition  is  necessary.  Suppose  P  =  QD,  where  D  =  o  has  all 
of  its  integrals  regular  and  Q  ^  o  has  no  regular  integral.  Every 
solution  oi  P  =:  o  which  satisfies  D  =  o  is  regular.     Every  solution 


420  LINEAR  DIFFERENTIAL   EQUATIONS. 

of  P  =  o  which  does  not  satisfy  D  ^  o  will  satisfy  one  of  the  equa- 
tions D  =  n,  where  7(  denotes  some  one  of  the  integrals  of  ^  =:  o.. 
Now  by  hypothesis  none  of  the  integrals  of  ^  =  o  are  regular,  and. 
if  in  D  we  replace  j/  by  a  regular  function,  the  result  is  a  regular  ex- 
pression, and  therefore  no  regular  value  oi  y  can  make  D  equal  to  u.. 
It  follows  then  that  D  =  7i  can  have  no  regular  solution,  and  conse- 
quently that  the  only  regular  integrals  oi  P  =  o  are  the  s  regular 
integrals  of  Z^  =  o. 

It  is  easy  to  see  the  cause  of  the  difference  y  —  s  between  the 
degree  y  of  the  indicial  equation  of  P  =  o,  and  the  number  s  of 
linearly  independent  regular  integrals  of  this  equation.  Suppose  in. 
fact  that  P  is  decomposed  into  prime  factors  of  the  form 

This  decomposition,  whatever  it  may  be,  will,  as  we  know,  contain 
y  regular  factors  ;  of  these  regular  factors  a  group  of  s  at  most  can 
be  placed  at  the  right-hand  end  of  the  decomposition,  and  the 
remaining  y  —  ^  factors  cannot  come  into  this  group  (of  course  in 
particular  cases  we  will  have  y  =  s).  The  difference  y  —  s  then 
arises  from  the  presence  of  regular  factors  in  Q  which  cannot  be 
placed  as  consecutive  factors  at  the  right-hand  end  of  the  decom- 
position of  Q. 

Observing  that  the  indicial  functions  of  D  and  Q  are  respectively 
of  the  degrees  s  and  y  —  s,  we  can  enunciate  the  two  following 
theorems : 

(a)  In  order  that  the  linear  differential  equation  P  ^^  o  of  order  n 
possessing  an  indieial  equation  of  order  y  shall  have  y  —  }x  li}iearly 
tndependetit  regular  integrals,  it  is  necessary  and  siifficicnt  that  P shall 
have  the  form  P  =  QD,  Q  and  D  being  of  the  same  form  as  P,  and 
that  Q=zo,  being  of  order  n  —  y  -\-  pi,  shall  have  no  regular  integrals 
and  shall  have  an  indicial  equation  of  order  pi. 

(/?)  In  order  that  the  linear  differential  equation  P  =.  o  of  order  n 
and  Jiaving  an  indicial  equation  of  degree  y  shall  have  y  linearly  in- 
dependent regular  integrals,  it  is  necessary  and  sufficient  that  P  shall' 
be  of  the  form  P  =  QD,  Q  and  D  being  of  the  same  form  as  P,  Q, 
being  of  order  n  —  y  and  having  a  constant  as  its  indicial  fit7ictio7U 


DECOMPOSITION  INTO   SYMBOLIC  PRIME  FACTORS.        42  I 

We  can  further  show  again  the  truth  of  the  theorem : 

{y)   The  s  linearly  independent  regular  integrals  ofP^=o  belong 

io  exponents  wJiich  are  certain  s  of  the  roots  of  the  indicial  equation 

for  P  z=  o. 

This  is  immediately  seen,  since  if  we  put  P  in  the  form 

P  =  QD, 
we  have  the  relation 

g{p)^  h{p)k{p-s) 

between  the  indicial  functions.  Now  the  s  regular  integrals  of 
jD  =  o,  that  is,  all  the  linearly  independent  regular  integrals  of 
P  =  o,  belong,  as  we  have  seen,  to  exponents  which  are  the  roots 
of  h{p)  =  o.  It  follows  then,  from  the  above  relation  connecting 
the  indicial  functions,  that  the  regular  integrals  of  /*  =  O  belong  to 
'exponents  which  are  certain  s  of  the  roots  oi  g{p)  =  o. 

The  following  theorems  concerning  the  adjoint  equation  are 
easily  seen  to  be  true ;  their  verification  will,  however,  be  left  as 
exercises. 

I.  The  indicial  functions  g{p)  and  Q{p)  of  the  adjoint  expressions 
P  and  p  are  derived  the  one  from  the  other  by  changing  p  into 
—  p  -\-  ^  —  I ,  where  x^  is  the  power  of  x  by  zvhich  tve  must  multiply 
P  in  order  to  put  it  in  the  normal  form. 

II.  If  the  linear  differential  equation  P  =  o  has  all  of  its  integrals 
regular,  the  same  is  true  of  its  adjoint  equation  |p  =  o. 

The  theorem  {y)  above  is  here  changed  into  the  following: 

III.  In  order  that  the  linear  differential  eqjiation  P  =  o  of  order  n 
having  an  indicial  equation  of  degree  y  shall  have  exactly  y  linearly  in- 
dependent regular  integrals,  it  is  necessary  and  sufficient  that  its  adjoint 
.equation  |p  ^  o  shall  have  among  its  integrals  all  the  integrals  of  a 
liiiear  differential  equation  of  order  n  —  y  which  has  a  constant  for  its 
indicial  finction. 

It  is  easy  to  see  from  the  preceding  investigations  that  all  the 
properties  of  the  regular  integrals  of  linear  differential  equations  can 
be  arrived  at  by  aid  of  the  decomposition  into  prime  symbolic 
;factors. 


CHAPTER   XI. 

APPLICATION    OF    THE    THEORY    OF    SUBSTITUTIONS    TO    LINEAR 
DIFFERENTIAL   EQUATIONS. 


We  may  recall  the  definition  of  the  product  of  two  substitu- 
tions ;  viz.,  the  product  SS^  of  the  two  substitutions  5  and  vSj  is  the 
new  substitution  which  produces  the  same  effect  as  if  the  substitu- 
tions 5  and  ^,  were  applied  successively.  A  similar  definition 
applies  to  the  product  of  any  number  of  substitutions.  In  particular, 
we  speak  of  the  power  of  a  substitution,  say  5" ,  which  means  that 
the  substitution  vS  has  been  made  a  times. 

In  Chapter  III,  equation  (2),  we  have  the  general  type  of  the 
substitution  which  corresponds  to  any  critical  point  of  the  linear 
differential  equation  with  uniform  coefificients ;  viz.,  if  j, ,  .  .  .  ,  ^» 
denote  a  system  of  fundamental  integrals  in  the  region  of  the 
assumed  critical  point,  we  have 


(I) 


5 


J«;       <^«  J'l  +  ^»2  7..  +     •     •     •     +    ^nn^n 

It  was  also  shown  how  by  aid  of  the  characteristic  equation- 


(2) 


^:u, 


<^2n 


■      ^H2 


=  o 


this  could  be  reduced  to  the  canonical  form  given  in  equation  (80')' 
of  the  same  chapter.  The  equation  (80')  may,  by  a  slight  change- 
in  notation,  be  written  in  the  form 

422. 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       423 


(3) 


5 


J/ 


y^^'\  -y.ji' 


j/  .  .  .  j/s  sy; ^ry.) 


yp 


s.(yp+/^-d 


^^: 


V^ ;   -y^^/  +  ^/) 


sXy /+//-.) 

2     1 


where  the  integers  k^  I,  .  .  .  satisfy  the  inequalities 

k^>  k^  .  .  .  -^  kk,     h>  K     •  •  •  > 

and  where  /^^  +  /^,  +  ...+  ^^^ ,  A  +  /,+  ...,...  are  the  re- 
spective degrees  of  multiplicity  of  the  roots  ^, ,  j-,  ,  .  .  .  of  equation  (2). 

We  shall  speak  of  the  functions  j,  -s^,  ...  as  forming  classes  cor- 
responding to  the  roots  j^ ,  j-,,  .  .  .  ,  respectively,  of  the  characteristic 
equation  ;  thus,  we  shall  speak  of  the  functions  j/  as  being  of  the  first 
class,  the  functions  z  as  being  of  the  second  class,  etc. 

The  group  of  the  differential  equation  has  been  already  defined, 
but  it  will  be  convenient  to  repeat  the  definition  here.  Suppose  a, 
b,  c,  .  .  .  to  be  the  critical  points  of  the  equation  ;  when  the  variable 


travels  round  any  one  of  these  points,  the  integrals  submit  to  a  sub- 
stitution of  the  form  (i),  or,  when  the  integrals  are  properly  chosen, 
of  the  canonical  form  (3). 

Any  path  going  round  only  one  critical  point  can  of  course  be 
replaced  by  the  loop  {lacet)  belonging  to  this  point,  and  any  path 


424  LINEAR  DIFFERENTIAL  EQUATIONS. 

going  round  more  than  one  of  these  points  may  be  replaced  by  the 
loops,  taken  in  proper  order,  belonging  to  these  points. 

Corresponding  to  each  of  these  paths  there  is  a  certain  substitu- 
tion ;  denote  by  A  the  substitution  corresponding  to  the  critical 
point  a,  by  B  the  substitution  corresponding  to  b,  etc.  If  in 
Fig.  I  we  start  from  O  and  travel  along  the  curved  line  back  to  O^ 
we  shall  have  gone  round  the  two  points  a  and  c.  Now  we  can  in 
the  usual  manner  (by  aid  of  the  dotted  line  OM)  see  that  this  path 
is  equivalent  to  the  two  loops  a  and  y  ;  we  therefore  apply  to  the 
functions  first  the  substitution  A  and  then  the  substitution  C,  or 
simply  the  substitution  AC. 

Again,  if  in  Fig.  2  we  start   from  O  and  go  round  the  curved 


Fig.  2. 
O 
line  and  come  back  to  O,  it  is  easy  to  see  that  we  have  applied  the 
substitution  ACE.     All  possible  substitutions  of  this  sort  that  we 
can  form  from  the  simple  substitutions  A,  B,  C,  .  .  .  ,  their  powers 
and  products,  form  the  group  G  of  the  equation. 

Denote  by  Fa  function-group  of  the  equation  such  that  all  the 
substitutions  of  G  transform  the  functions  j,  ^,  it,  .  .  .  among  them- 
selves. It  is  clear  that  this  function-group  can  be  regarded  as  con- 
taining certain  other  function-groups.  For  example,  suppose  we 
consider  the  single  function  y  and  its  transforms  by  A  and  the 
powers  of  A  ;  these  will  form  a  function-group  which  is  obviously 
contained  in  F;  and  so  in  many  different  ways  we  may  form  func- 
tion-groups which  will  all  be  cojitaincd  in  F.  We  will  call  such  func- 
tion-groups minors  oi  F. 

We  will  show  now  that  F  can  be  considered  as  derived  from  a 
certain  number  of  distinct  functions,  each  of  which  contains  only 


APPLICATION  OF    THE    THEORY  OF  SUBSTITUTIONS.       425 

A^ariables  of  a  single  class.  Any  function  whatever  in  F  is  made  up 
of  the  sum  of  a  certain  linear  function  of  the  variables  j  (say  F),  a 
linear  function  of  the  variables  s  (say  Z),  a  linear  function  of  the 
variables  71  (say  U\  etc.  If  then/ denote  a  function  contained  in  F, 
we  can  write 

(4)  /-F+Z+^+ 

Let  0  denote  the  function-group  derived  from  /and  its  trans- 
forms f\  f",  .  .  .  which  are  obtained  by  operating  on  /  by  the 
•different  powers  of  A  ;  again,  let  ^'  denote  the  function-group  formed 
from  the  partial  functions  V,  Z,  .  .  .  ,  and  their  transforms  by  A, 
A',  ....      We  may  write 

(5)  0'=^',-^^\+  .  .  .  ; 

0'y  being  made  up  of  y  and  its  transforms,  ^\  of  Z  and  its  trans- 
forms, etc.  It  is  obvious  that  the  function-group  ^  is  a  minor  of  ^'. 
Let  now  p  denote  the  greatest  of  the  indices  k  of  those  of  the  func- 
tions j*  which  enter  in  F;  cr  the  greatest  of  the  indices  /  of  those  of 
the  functions  ^-  which  enter  in  Z;  etc.  We  can  now  establish  the 
following  theorem : 

T/ie  function-group  ^  contains  each  of  the  partial  functions  F, 
Z,  .  .  .  and  is  derived  from  Q -^  (T  A^  .  .  .  distinct  functions,  of  tvhich 
p  are  formed  exclusively  with  the  variables  y],  a  are  formed  exclu- 
sively with  the  variables  z\ ,  etc. 

We  have  to  show  first  that  the  number  of  distinct  functions  in 
<^  is  not  greater  than  p  -j-  o-  -f-  .  .  . 

We  will  determine  first  how  many  distinct  functions  will  be  given 
by  F  and  its  transforms.  Denote  by  Fp_,-  a  linear  function  of  the  /s 
whose  index  is  less  than  or  equal  to  p  —  i.  Now  it  is  obvious  that 
^'  contains  the  function 

(6)  I^=^V+A'V+  .  .  .  +F,_,, 

and  it  also  contains  the  transform  A  V,  and  consequently  it  contains 
the  function 

(7)  V  =  ~-AV-V, 
which  has  obviously  the  form 

<8)  y  -  A>;_,  +  A-y',_,  +  .  .  .  +  F,_.. 


426  LINEAR  DIFFERENTIAL  EQUATIONS. 

The  functions  Y  and  Y'  are  by  definition  elements  of  the  sub- 
function-group  ^' y .  By  a  continuation  of  the  above  process  we  see 
that  0'y  contains  the  elements 

(9)  r'  =  --AY-Y  =  A>',_,  +  A'y  V.  +  .  .  .  +  F,_3 ; 


(lo)     F"-'  =  ^-A  yp--  -  yp-  =  A>/  +  ry,"  +  .  .  .  . 

This  last  function  gives  identically 

(i  i)  -   —A  Fp-'  —  Fp-'  =  o. 

The  functions  Y,  Y',  .  .  .  ,  Yf-^  are  evidently  linearly  independent, 
as  each  of  them  contains  certain  of  the  variables  y  which  are  not 
contained  in  the  others.     Again,  we  have  from  the  above  equations 

(12)  AY=s,{Y-{-  Y'),  AY'  =s{Y'-\-  Y"\  .  .  .  ,  AY"-' =  s,Yp-^  \ 

that  is,  the  substitution  A  changes  these  functions  into  linear  func- 
tions of  themselves  alone.  The  different  powers  of  ^  will  obviously 
do  the  same  thing,  and  consequently  the  number  of  linearly  inde- 
pendent functions  in  ^'y  is  exactly  equal  to  p.  In  the  same  way 
we  see  that  the  number  of  distinct  functions  in  ^\  is  equal  to  a,  and 
so  for  the  other  sub-function-groups.  It  follows  then  that  0'  con- 
tains exactly  p  -\-  a  -{-  .  .  .  distinct  functions,  and  consequently  that 
0,  which  is  a  minor  of  0',  can  contain  no  more  than  p  -\-  a  -\-  .  .  . 
distinct  functions.  To  complete  the  proof  of  the  above  theorem, 
it  is  now  necessary  to  show  that  0  contains  at  least  p  -f-  cr  -j-  .  .  . 
linearly  independent  functions. 

It  is  obvious  that  among  the  functions  composing  0  we  have 

(13)  /=Aj,  +  Ay,^-  .  .  .  +F,_,  +  Z+f/-f  .  .  .  ; 

Z  being  a  linear  function  of  the  variables  s,  U  a.  linear  function  of 
the  variables  ?/,  etc.  From  tli.e  definition  of  0  we  see  that  this  func- 
tion-group contains  A/,  A'/,  ...  It  follows  then  that  ^  contains 
also  the  function 

W-sJ)  =  Ajv  +  A'/p  +  .  .  .  +F;_.  +Z'+  .  .  . 


APPLICATION  OF    THE    THEORY  OF  SUBSTITUTIONS.       42/- 

Y' p^.^  is  analogous  to  Fp_i,  and  Z'  is  analogous  to  Z,  but  differs 
from  Z  in  that  the  highest  index  of  the  variables  z  is  now  equal  to 
or  less  than  o-  —  i.     In  like  manner  we  have  in  ^  the  function 

(15)   f"  =  Y^W  -sj') 

=  Aj/,  +  Ay,+  .  .  .  +  F"p_.  +  Z-  +  .  .  .   ; 

the  highest  index  of  s  in  Z"  being  equal  to  or  less  than  cr  —  2,  etc. 

We  can  obviously  continue  this  process  until  we  arrive  at  a 
function  contained  in  0  and  containing  none  of  the  variables  s. 
Suppose  now  that  the  variables  u  belong  to  the  root  s^  of  the  char- 
acteristic equation;  then  by  a  process  entirely  similar  to  the  preced- 
ing we  can  find  a  function  contained  in  <?  which  contains  none  of  thp 
variables  7i,  and  so  for  all  the  other  variables  except  those  of  the 
first  class.  We  have  then  finally,  as  one  of  the  functions  in  0,  the 
expression 
(i6)  0  =  Ajp  +  A>/+  .  .  .  -fll)p_,; 

3^p_i  being  a  linear  function  of  the  variables  j/  whose  index  is  less 
than  p. 

The  functions  A(p,  A^(p,  .  .  .  are  all  contained  in  0,  and  of  the 
set  (p,  A(p,  A^(p,  .  .  .  there  are  evidently  p  distinct  functions.  In 
the  same  way  we  can  show  that^  0  contains  (X  distinct  functions 
involving  only  the  variables  s,  etc. — that  is,  0  contains  in  all 
p  -\-  (T  -{-  .  .  .  distinct  functions ;  and  since  ^  is  a  minor  of  0',  which 
contains  the  same  number  of  distinct  functions,  it  follows  that  0  and 
0'  are  identical,  and  therefore  0  contains  the  functions  V,  Z,  17,  .  .  . 
above  defined.  Now  (^  is  a  minor  of  the  function-group  F;  there- 
fore F  contains  V,  Z,  .  .  .  among  its  elements.  We  have  then, 
finally,  that  if  the  function-group  F  contains  the  function 

(4)  /=F  +  Z+t/4-  .  .  ., 

it  contains  also  the  functions  V,  Z,  U,  .  .  . 

Let  now  F, ,  V^,  .  .  .  denote  the  linearly  independent  functions 
contained  in  i^  which  contain  only  the  variables  y ;  Z^,  Z^,  .  .  .  the 
analogous  functions  of  the  variables  s  ;  etc.  The  function-group  F  is. 
derived  irovci  the  elements  Fj ,  F^,  .  .  .  ,  Z^,  Z^,  .  .  .  ,  etc.  For,  de- 
noting by 

/,=  V+Z-i-  .  .  ., 


428 


LINEAR   DIFFERENTIAL   EQUATIONS. 


any  one  of  the  functions  contained  in  F,  we  know  that  F,  which 
depends  only  on  the  variables  jj',  is  a  linear  function  of  F, ,  K,,  .  .  .  , 
and  that  Z,  containing  only  the  variables  z,  is  a  linear  function  of  \ 
Z^,  Z^,  .  .  .  ,  etc.  We  have  thus  established  the  above  proposition  J 
concerning  the  function-group  F,  viz. :  that  F  can  be  considered  as  de-  -. 
rived  from  a  certain  number  of  distinct  functions  F, ,  F^ ,  .  .  .  ,  -2", , 
Z^^  .  .  .  ,  etc.,  each  of  which  contains  only  variables  of  a  single  class. 
Suppose  that  in  F  we  have  the  function  F  of  the  variables  y,  say 


(17) 


F,=/,  =  A>/  +  A'>/'+. 


The  different  transforms  of  /  are  also  contained  in  F.  If  we  per- 
form the  substitution  A  any  number  of  times  in  F,  we  obtain  func- 
tions which  only  contain  the  variables/;  but  if  we  perform  the  sub- 
stitutions B,  C,  etc.,  we  get  new  functions  which  are  linear  on  the 
one  hand  with  respect  to  the  variables  y,  s,  ti,  .  .  .  ,  and  on  the  other 
hand  with,  respect  to  the  constants  A',  A'',  .  .  .  Denote  by/*,  ,/*, ,  .  .  . 
the  different  products  that  can  be  formed  by  multiplying  one  of  the 
constants  A  by  one  of  the  variables  y,  z,u,  .  .  .  ;  as  both  variables  and 
constants  are  finite  in  number,  there  will  be  only  a  finite  number  of 
the  products  P.  Let  tj,,  r/^,  •  .  •  denote  linear  functions  of  the  y's 
alone,  C,>  C,?  •  •  •  linear  functions  of  the  ^•'s  alone,  etc.;  then  we  can 
write  • 

(18)  ^/=7>  +  C,+  .  .  ., 


(19) 


Cf  =  V.  +  c.  + 


From  what  we  have  above  proved  it  is  clear  now  that  F  contains 
ih^  7,.  •  •  .  ,  C,  C  •  .   •  ,  and  also 


^o) 


/  /I  ///J/  / 


i-^c.-c,  c/'  =jAc:-Q/ . . ., 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       429 

each  of  which  contains  only  variables  of  a  single  class.  These  dif- 
ferent functions  ?;, ,  ;;/,  ?//',  .  .  .  ,  Ci ,  C/j  •  •  •  are  of  course  linear  func- 
tions oi  P^,  P^,  .  .  .  It  may  be,  however,  that  some  of  these  functions 
77,  C,  •  •  •  ,  7',  Z',  •  ■  •  are  linearly  expressible  in  terms  of  the  remain- 
ing ones  and/";  if  such  functions  exist  we  will  discard  them,  and  so 
have  finally  a  certain  set  of  distinct  functions,  say/",/',  f",  .  .  . ,  /% 
of  which  /'.../''  have  been  derived  by  operating  upon  /  by  B, 
C,  .  .  .  and  by  powers  of  A.  We  may  take  now  each  of  these  new 
functions  /',  f",  .  .  .  ,/'',  and  by  precisely  similar  operations  arrive 
at  still  other  functions  which  will  be  contained  in  P  and  each  of 
which  will  contain  only  the  variables  of  a  single  class.  These  new 
functions  are  again  linear  functions  of  P^,P^,  .  .  .  ;  neglect,  as  be- 
fore, all  of  them  that  are  linear  functions  of  the  remaining  ones  and. 
of  /,  /',  .  .  .,/'',  and  say  the  new  set  of  distinct  functions  is 

Treat  the  functions  /''+'  ...  /"^  as  before,  and  neglect  all  the  new- 
functions  which  are  linearly  expressible  in  terms  of  the  remaining 
ones  and  of  /",  /',  .  •  .  ,  /^  We  will  continue  this  process  until  we 
can  find  no  new  function  which  is  not  linearly  expressible  in  terms 
of  those  already  found.  That  the  series  of  operations  is  limited  is 
clear  from  the  fact  that  each  of  the  functions  which  we  form  is  a  linear 
function  of  the  quantities  P^,  P^,  .  .  .  which  are  limited  in  number. 
Suppose  the  functions  finally  obtained  are  /,  /',  .  .  .  ,  /S  each  of 
which  contains  only  the  variables  of  a  single  class  (of  course  there 
will  in  general  be  several  functions  which  contain  the  variables  of 
the  same  class).  Each  of  the  functions/,  /',  .  .  .  ,  /'^  is  transformed 
by  each  of  the  substitutions  of  the  group  G  into  a  linear  function  of 
/)  /'>  •  •  •  >  /'•  In  order  to  prove  this  it  is  only  necessary  to  show 
its  truth  for  each  of  the  substitutions  A,  B,  C,  .  .  .  from  which  G  is 
derived.  Take,  for  example,  the  function  /',  =  ?//.  The  transform, 
of  this  by  the  substitution  A  is 

(21)  A,f/  =  sXvr  +  v/); 

operating  with  B,  C,  .  .  .  ,  we  have 

(22)  Brj,'  =  ^7,  +  c;  +  .  .  .  , 

(23)  C7j/  =  V.  +  ^.+  .  .  .  .• 


430 


LINEAR  DIFFERENTIAL  EQUATIONS. 


I 


//,  Q,  .  .  .  containing  respectively  only  the  variables  j,  ^,  .  .  .  ;  but  ^ 
each  of  these  partial  functions  /;,  C>  •  •  •  is  a  linear  function  of  /, 
/\  .  ,  .  ,  p .  From  the  functions  /",  /"',  .  .  .  ,  p  we  can  form  a  func- 
tion-group W  which  possesses  the  property  of  having  its  different 
functions  transformed  into  one  another  by  the  substitutions  of  the 
group  G. 

Suppose/, /j  j/s,  ...  to  be  the  functions  which  depend  only  on 
the  variables  J,  and  let  Wy  be  the  function-group  derived  from/,/ , 
y, ,  .  .  .  ,  or,  what  amounts  to  the  same  thing,  derived  from  /  0, , 
03,  ...  ,  where 


(24) 


01  =  /i  +  ^i/     0.  =  /.  +  ^./ 


Ci?! ,  gl>2  ,  .  .  .  being  constants  each  of  which  is  subject  to  only  one 
condition,  viz.,  that  it  shall  not  be  a  root  of  a  certain  algebraic  equa- 
tion.    The  function /is  of  the  form 

(25)  /=A>/-f  r>/'+  .  . . , 

and  /  of  the  form 


(26)  /  =  \_ay:  -f  by:'  +  .  .  .  +^j../  +  .  .  .  ]A' 

therefore  0,  is  of  the  form 

(27)  0,  =  \{a  +  oo:)y:  +  by:'  +  .  .  .  +  ^^/  +  , 


F'  + 


We  will  now  determine  (»,  in  such  a  manner  that  the  determinant 

a  A-  00^,     b  ,     d 

a  ,'    b'  -\-  Qo^i     d' 

a'  ,     b"  ,     d"  +  Go^  .  . 


(28) 


A  = 


shall  not  vanish.  It  is  obviously  only  necessary  to  choose  for  g?,  any 
value  that  is  not  a  root  of  the  equation  Jj  =  o.  The  constants  go,  , 
co^,  .  .  .  are  to  be  determined  in  a  similar  manner;  that  is,  a?,  must 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       43 1 

-not  be  a  root  of  a  certain  equation  A^  =  o,  etc.     The  function  <p  is 
therefore  obtained  by  operating  upon 


/=A>/  +  r>/'+  . . . 


with  the  substitution 


2.= 


y:  ;     {a  +  a.,>/  +        by,"        + 


The  functions  cp^,  cf)^,  .  .  .  will  be  obtained  by  operating  upon /with 
similar  substitutions  2^,  2^,  .  .  .  .  It  follows,  therefore,  that  the 
function-group  Wy  will  be  identical  with  the  function-group  derived 
from  the  transforms  of /by  the  substitutions  of  a  substitution-group 
F  which  is  derived  from  2,,  2^,  .  .  .  Let  us  assume  now  that  we  are 
able  to  ascertain  whether  the  group  F  is  prime  or  not,  and  in  this 
second  case  we  will  suppose  that  we  can  determine  a  function-group 
©which  is  derived  from  a  number  of  distinct  functions,  this  number 
being  less  than  the  number  of  the  variables  y ;  further,  Q  is  to  be 
such  that  the  substitutions  2, ,  ^^^  •  •  •  shall  replace  its  functions 
the  one  by  another.  Suppose,  first,  that  F  is  not  prime,  and 
•denote  by  ^  any  one  of  the  functions  in  0.  Choose  the  constants 
X',  X",  .  .  .  ,  such  that 


<29) 


^.-A>/  +  r>/'  + 


The  function  ^  and  all  of  its  transforms  by  the  substitutions  ^, , 
^2,  .  .  .  will  form  a  function-group  contained  in  0,  and  consequently 
depending  upon  a  number,  say  p,  of  distinct  functions,  at  most  equal 
to  the  number  of  distinct  functions  in  0.  Now  among  the  distinct 
functions  from  which  W  is  derived,  and  each  of  which  contains  only 
one  class  of  variables,  there  will  be  only  p  containing  the  variables 
y,  and  the  number  of  these  variables  is  by  hypothesis  greater  than  p. 
In  the  case  of  each  of  the  other  classes  there  can  of  course  be  no 
more  functions  than  the  number  of  variables  of  the  corresponding 
class.  The  function-group  W  will  then  contain  less  than  7t  distinct 
functions,  which  will  be  obtained  by  substituting  in  /  /,  .  .  .  ,  /'  the 
special  values  of  X',  X" ,  .  .  .  and,  after  effecting  the  substitutions, 


432  LINEAR  DIFFERENTIAL   EQUATIONS. 

neglecting  those  functions  which  are  Hnearly  expressible  in  terms  of" 
the  others. 

In  this  case,  then,  the  group  G  is  not  prime,  and  we  have  deter- 
mined a  function-group  containing  less  than  n  distinct  functions. 

In  the  second  case,  suppose  F  to  be  prime.  Whatever  be  the 
system  of  values  chosen  for  A',  A",  .  .  .  ,  the  function-group  Wy  ob- 
tained from  \'y^  -\-  \"y^'  -1-  ...  by  the  substitutions  of  the  group 
7^  will  depend  upon  a  number  of  distinct  functions  equal  to  the 
number  of  the  variables  j.  It  follows  then  that  Wy  will  contain 
among  its  functions  all  those  that  can  be  formed  from  these  varia- 
bles. Suppose  j/  one  of  these  functions ;  then  yl  is  necessarily 
contained  in  F.  Since  A',  \" ,  .  .  .  are  no  longer  restricted  in  value, 
we  may  write  A'  =  i.  A"  =  'K!"  =  .  .  .  =  o,  and  ascertain  whether 
or  not  the  function-group  W  formed  under  this  hypothesis  contains 
n  or  less  than  n  distinct  functions.  If  W  contains  n  distinct  func- 
tions, then  F,  of  which  ^  is  a  minor,  will  also  contain  n  distinct 
functions,  and  G  will  therefore  be  prime.  If  W  contains  less  than  n 
distinct  functions,  G  will  not  be  prime,  and  the  system  of  distinct 
functions  upon  which  W  depends  will  have  to  be  determined. 

Suppose  now  that  /^contains  no  function  of  the  variables  y  of 
the  first  class,  but  contains  a  function  Z  of  the  variables  z  of  the 
second  class.  By  a  procedure  identically  the  same  as  the  one  just 
described  we  can  determine  a  function-group  W  (if  such  exists) 
which  shall  be  a  minor  of  i^  depending  upon  less  than  n  distinct 
functions.  And  so  in  like  manner  we  can  proceed  if  F  contains  no 
function  of  the  variables  y  and  no  function  of  the  variables  z,  but 
a  function  ^of  the  variables  u  of  the  third  class,  etc.  By  continu- 
ing this  process  we  see  that  we  can  always  determine  a  function- 
group  W  containing  less  than  n  distinct  functions,  unless  indeed  F 
is  known  to  contain  always  n  distinct  functions,  in  which  case  G 
is  prime  by  definition. 

The  question  proposed  in  Chapter  VIII  concerning  the  group  G 
was,  "  Having  givcji  a  group  G  composed  of  linear  substitutions 
A,  B,  C,  .  .  .  among  n  variables,  required  to  determine  zuhether  or  not 
the  linear  differential  eqiiation  zvhicJi  has  G  for  its  group  is  satisfied  by 
the  integrals  of  analogous  differential  equations  of  orders  lower  than  riy 
and  to  determine  t  lie  groups  of  these  equations'' 

In  what  precedes  we  have  supposed  that  we  knew  how  to  answer 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       433 

this  question  for  the  group  F  and  all  similar  groups.  The  number 
of  variables  in  F  is  less  than  those  in  G  (since  F  only  contains  the 
variables  of  a  single  class),  and  the  problem  is  therefore  reduced  to 
a  simpler  form,  and  may  be  considered  as  solved  when  we  examine 
the  case  so  far  excluded;  viz.,  the  case  when  the  characteristic  equa- 
tion corresponding  to  the  substitution  .S  has  all  of  its  roots  equal — 
or,  say,  has  but  one  root. 

We  will  suppose  (changing  very  slightly  the  previous  notation) 
that  the  characteristic  equation  corresponding  to  the  substitution  A 
has  the  single  root  a.  In  order  that  we  may  have  a  new  problem  to 
solve  it  is  necessary  that  the  characteristic  equation  corresponding 
to  each  and  every  substitution  in  G  shall  have  but  one  root ;  if,  for 
example,  there  existed  in  6^  a  substitution,  say  S,  corresponding 
to  which  there  was  more  than  one  root,  then  we  might  reason  with 
^  as  we  have  already  with  A  and  effect  the  required  reduction  by 
the  processes  described  above.  We  must  therefore  suppose  that 
there  is  no  substitution  in  G  whose  equation  has  more  than  one  root. 
We  will,  in  this  case,  first  show  that  G  cannot  be  prime,  and  will 
give  the  canonical  form  of  its  substitutions  which  shall  bring  this  fact 
into  evidence. 

This  being  done,  we  will  show  how  to  ascertain  whether  or  not 
the  assumed  hypothesis  is  admissible,  and,  if  it  is,  give  the  canonical 
form  of  the  substitutions  of  G.  In  case  the  hypothesis  is  not  cor- 
rect we  will  show  either  how  to  obtain  directly  a  function-group 
containing  less  than  n  distinct  functions,  or  a  substitution,  say  5, 
corresponding  to  which  the  characteristic  equation  has  several  dis- 
tinct roots,  and  so  be  conducted  to  the  case  already  investigated. 

Denote  by  a,  l\  .  .  .  the  single  roots  of  the  characteristic  equa- 
tions corresponding  to  the  substitutions  A,  B,  .  .  .     We  may  write 

A  =aU,     B  =  d:B,     ... 

Here  a,  d,  .  .  .  represent  substitutions  which  multiply  all  the  vari- 
ables by  a,  d,  .  .  .  ;  and  H,  JS,  •  •  •  substitutions  whose  determinant 
is  unity  and  corresponding  to  each  of  which  we  have  the  character- 
istic equation 

(s  —  i)"  =  o. 


434 


LINEAR   DIFFERENTIAL   EQUATIONS. 


We  have  now  a  group  0  derived  from  H,  !B,  •  •  •  as  G^  is  derived 
from  A,  B,  .  .  .     Consider  any  substitution,  say  T,  of  G,  and  let 


T=  a  b^ 


H-^JS^ 


then 


and  this  is  a  substitution  contained  in  Q>.     Let 

A  —  5,     /< 

A,  ,  /<j  —  J  .  .  . 


(30) 


A  = 


=  o 


be  the  characteristic  equation  corresponding  to  TT.  The  roots  of  the 
characteristic  equation  corresponding  to  7"  are  evidently  the  roots 
of  zJ  =  O  multiplied  by  the  constant  {a'^d^  ...)";  but  by  hypothesis 
this  last  equation  has  only  one  root,  therefore  z/  =  o  has  only  one 
root. 

We  will  now  show  that  ^/le  single  root  of  A  =^  o  is  unity.  In  order 
to  do  this,  we  will  show  that  if  this  statement  is  true  for  any  two 
substitutions  S  and  XT,  it  is  true  for  their  product  SXT  ;  then  as  we 
know  that  the  proposition  is  true  for  the  substitutions  H,  iB,  •  •  •  , 
from  which  (5  is  derived,  it  must  also  be  true  for  their  products 
taken  two  and  two,  three  and  three,  etc.,  and  consequently  for  all 
the  substitutions  of  (5.  , 

We  may  evidently  suppose  the  variables  y,  z,  .  .  .  so  chosen  that 
the  substitution  S  shall  be  in  the  canonical  form ;  viz.,  by  the  above 
hypothesis. 


(30 


S  = 


Ji •  •  •  a;  y^'  '-yr 

z,...z,;     z^^y,...z,-^y, 
u^.  .  .Ut\     n,-\-  z^  .  .  .nt-\-  Zt 


The  substitution  XT  cannot,  of  course  (with  these  variables),  be  sup- 
posed  to  be  in  its  canonical  form,  so  we  may  write 


(32) 


U  = 


J: ;     a,y,  +  .  .  .  +  b.-.  +  •  •  •  +  ^i^^  +  •  •  • 

?^ ;    < j>  +  •  •  •  +  ^>-i  +  •  •  •  +  /?^  +  •  •  • 


APPLICATION  OF    THE    THEORY  OF  SUBSTITUTIONS.       435 
From  these  last  two  equations  we  have  (A  denoting  an  integer) 


(33)      S^XT 


A(A  -  I) 


J. ;    Ui  +  ^'^i  +  -y- 


^. +  . 


J'.  +  .. 


+(^,+Ar,+  . .  .K+ . . .  +(^.+  .  •  •>.+ 


«i; 


'^,  +  ^'',  +  '^^V.+...i/.+ 


I  .  2 


+(^:+A/,+   .   .    .K+    •   •    •  +(/+  •   .    •>:+   . 


The  characteristic  equation  corresponding  to  this  substitution  may 
be  written  in  the  form 


(34) 


J  =  J"  +  0^"-'  +  0/'-^  4-  .  .  .  _|-  (_  i)«/)  —  o, 


where  D  is  the  determinant  of  SXT,  and  where  0,  0^ ,  ,  .  .  are  inte- 
gral functions  ;  but  since  5  and  TI  are  products  formed  with  the 
substitutions  H,  !B,  ...  each  of  whose  determinants  is  unity,  it  fol- 
lows that  D  =  \.  The  single  root  of  J  =  o  will  then  be  an  n^^  root 
of  unity,  since  the  n^^  power  of  this  root  is  (to  sign  pres)  the  last 
term  of  equation  (32) — i.e.,  unity.     We  have  then 


(35) 


A  =  {s—  6)"  =  o, 


where  0  is  an  n'-^  root  of  unity.     By  comparing  (34)  and  (35),  we  see 
that  we  must  have 


(36)  0  = 

and  consequently 


7l(?l  —  i)    , 


1 .  2 


(37)  0"  =  (-  ^^r,   <t>r = I  \  ^  _, 

These  equations  are  of  finite  degree  in  A  and  are  satisfied  by 
hypothesis  for  all  values  of  A ;  they  are  therefore  identities.  The 
coefificients  0,  0^ ,  .  .  .  are  therefore  constants,  and  z/  =  o  is  inde- 
pendent of  A.  Suppose  now  A  =  o;  then  by  hypothesis  again  we 
have    J  =  {s  —  ly.      It    follows    then    that    all    the    substitutions 


436 


LINEAR  DIFFERENTIAL   EQUATIONS. 


SXT  .  .  .  S^tl  have  for  the  corresponding  characteristic  equatiom 
simply  {s  —  i)"  ^  O,  and  so  finally  every  substitution  in  (3  has  for  it^ 
characteristic  equation  {s  —  i)"  =  o. 

We  will  now  suppose  the  variables  so  chosen  that  the  substitu- 
tion H  is  in  its  canonical  form,  and  for  simplicity  we  will  assume 


(38) 


H  = 


j'l '  y-i .  y^ ,  y. ;  yi ,  y^ .  y, ,  y. 

Any  other  substitution,  say  XT,  of  0  may  be  written  in  the  form 


(39)     ^=-- 


j'p ;    «pO'i  +  •  •  •  +  <h^y,-\-  ^9.-.  +  •  •  •  +'^P3-^ 

2p  ;        ^Pi  Ji     +    •    •    •    +   ^P4/4   +   ^Pi-.  +    •    •    •  +«'P3'^3 


We  have  now 
(40)     %Ji  = 

yp-,    (^pi+^'^px)j^+ 
-p ;    (^pi  +^«'pi)ji+ 


•    •   +  (^P3+^^P3)/3  +  ^P4  74+^Pi-i  + 
•    •    +   (^P3+'^^P3)J^3  +  ^P474+^P'-i  + 


The  characteristic  equation  corresponding  to  this  is  obviously 
^n  +  ^^u  —  s,     a,,  +  Xb,„_ 


(41) 


=  o, 


which,  as  already  shown,  is  independent  of  A,    In  expanding  this  we 
find  certain  terms,  viz.,  those  contained  in  the  expression 


(42) 


A/^2, ,  A<^j3  —  s,      A<^„ 

^^^31  >  A^3j ,  A(^33  —  s 


s\ 


APPLICATION  OF    THE    THEORY  OF  SUBSTITUTIONS. 


437 


involving  only  the  constants  b^.^,  b^^,  .  .  .  ,  b^^  and  not  cancelling 
with  any  other  terms.  The  terms  containing  these  constants  b  must 
therefore  vanish  identically,  and  so,  equating  to  zero  the  coefficients 
of  A/,  X^s^,  A.V,  we  must  have 


(43) 

b 

:  +   ^^22  + 

<^33    = 

=    0; 

(44) 

K, 
Ki  y 

<^22 

+ 

"22  '       "23 

^^32    »          ^33 

+ 

^^33    » 

^n  .        ^12  .         ^:3 

(45) 

^^21    .          ^22   »           '^23 

=    0. 

b 

Jl    J          ^32   ' 

^33! 

=  0; 


We  may  remark  here  that  since  we  have  only  the  three  independent 
variables  s^  ,  z^,  s^ ,  we  may,  without  altering  the  substitution  H, 
replace  j^ ,  j/^ ,  j,  by  arbitrary  linear  functions  of  themselves,  pro 
vided  that  we  make  an  analogous  change  in  ^, ,  z„,  z^.  By  aid  of 
this  very  obvious  remark  we  may  obtain  a  simpler  form  for  the  sub- 
stitution XT- 

From  (45)  we  can  clearly  determine  three  constants,  say  /, ,  4,  /g, 
such  that 

A('^n-,  +  ^,2-3'2  +  '^n^a) 

(46)  +/2(^2,--:+'^.2^-2+'^23^-3) 

+  ^3('^3,-l  +  ^32'3'2  +    -^33-3)     =    O. 

Supposing  then  that  /^ ,  4  ,  /g  are  determined  so  as  to  satisfy  (46)  ;  if 
110W  we  replace  one  of  the  variables y^,  y^,  y^  by 

it  is  easy  to  see  that  the  corresponding  transform  of  this  variable 
in  the  substitution  tl  will  not  contain  z  ;  if  j,  be  the  variable  so 
replaced,  the  effect  in  1^  will  be  the  same  as  if  the  coefficients 
b-ix ,  <^32  J  ^M  were  made  zero.  Suppose  then  that  this  operation  has 
been  performed  and  that  we  have 


(47) 


^^33  =  0; 


438 


LINEAR  DIFFERENTIAL  EQUATIONS. 


it  follows  now  that  (z^)  reduces  to 

b.,  ,     b, 
(48) 


=  o. 


By  aid  of  this  equation  we  can   determine  two  constants  m^ 
such  that 


m„ 


(49) 


wX'^n-i  +  b,„z^  +  mlp^.z^  +  b^^z^  —  o  ; 


if  now  we  replace  j'^  by  wj'i  -f-  vi„_y^ ,  we  will  not  affect  the  substitu- 
tion but  H,  we  will  produce  in  '^  the  same  result  as  if  we  made 
^21  =  ^11  =  O-  This  change  effected,  reduces  (43)  to  b^^  =  o.  We 
see  then,  finally,  that  we  may  write 


(50) 


b,,  =  b^^  =  b. 


b,,  =  k„  =  b. 


Further,  it  is  easy  to  show  that  if  one  of  the  coefficients  b^^ ,  b^^  is 
zero,  we  may  also  make  the  other  zero. 

(i)  Let  b^^  =  o:  we  can  obviously  find  /j  and  4,  such  that 
/j<^i3  -\-  4^23  =  6.  If  <^,3  is  not  zero,  then  /,  cannot  be  zero,  and  we 
may  replace  j/^  and  s^  by  /jji  +  4^2  ^^^  K^\  ~h  4-2  >  which  will  not 
affect  H,  but  will  make  the  resulting  coefficient  to  XI  which  is  analo- 
gous to  the  original  b^^  equal  to  zero.  If  b^^  =  O,  we  shall  arrive  at 
the  same  result  by  permuting  the  variables  }\  ,  ^,  with  y^,  z^. 

(2)  Suppose  b^^  =  o:  if  b^^  is  not  zero,  we  can  cause  the  new 
coefficient  in  XL  which  is  analogous  to  the  original  b^^  to  vanish  by 
taking  for  new  variables  ^,„ji/„  -|-  b^^j^^,  b^^z^  -\-  b^^z^  in  place  of  jf^  and 
^3 ;  if  <^,3  =  o,  the  same  result  will  be  obtained  by  permuting  jjj,  z^ 
with  ji/3 ,  5-, .  I 

Denote  now  by 


(51)  m  = 


any  substitution  of  the  group  (3  ;  write,  for  brevity, 

(52)     b"p^  =  {a'p,  -f-  /ub'p,)b,^  +  (rt'p,  +  /.ib'p^y^^  -\-  {a'p^  -\-  fxb'p^b^^ 

+  i'i'pJU'^  +  ^'pi^KT  +  b'pj^^  +  b'p.d^^ ,  etc., 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       439 

where  p.  is  an  indeterminate  constant.     The  substitution  XTH'^'Gl, 
=  It),  will  be  of  the  form 


(53)     D  = 


As  already  shown,  the  coefificients  d^/',  .  .  .   ,  dj'  must  satisfy  equa- 


tions of  the  same  form  as  (43),  (44),  and  (45),  and  in  particular 

(54)  ^„"  +  V'  +  ^33"  =  o. 

Substituting  here  the  values  of  d,,",  dj',  and  I? J'  from  (52),  and 
equating  to  zero  the  coefficients  of  pi,  we  have,  on  taking  account  of 
equations  (50), 

(55)  ^'J\.  +  '^'3i^>3  +  '^'3./'.3  =  o. 

This  equation  must  necessarily  be  satisfied  whatever  be  the  substi- 
tution XH  of  the  group  (3  ;  it  must  therefore  be  satisfied  when  we 
replace  the  coefficients  d\, ,  b\^ ,  b\^  of  XOl  by  the  corresponding 
coefficients  bj',  bj',  bj'  of  H) ;  consequently,  whatever  may  be  the 
value  of  /i,  we  must  have 

(56)  <^./'^.+    ^3/'^.3   +    '^3/'^.3    =    0. 

If  now  in  this  equation  we  replace  bj\  bj\  bj'  by  their  values 
and  equate  to  zero  the  terms  multiplied  by  jj.,  we  shall  have,  by  aid 
of  (50), 

(57)  ^JJ\^  =  O. 
This  equation  can  be  satisfied  in  two  different  ways :  first,  by  making 

(58)  b\,=o- 

second,  by  making  b^J?^^  =  o.  But  we  have  shown  that  if  either  /^^^ 
or  b„^  is  zero,  the  other  may  also  be  made  zero,  and  so  the  equation 
^lAs  =  o  is  equivalent  to 

{59)  ^12  =  o,     b,,  =  o, 


440  LINEAR  DIFFERENTIAL  EQUATIONS. 

and  finally,  from  (55), 

(60)  b,,  =  o. 

It  is  now  easy  to  establish  the  first  of  our  theorems,  viz.,  that 
under  the  assumed  hypothesis  as  to  the  roots  of  the  characteristic 
equations  corresponding  to  the  substitutions  of  the  group  (3,  this 
group  cannot  be  prime.  There  are  two  cases  to  be  considered 
according  as  d\^  is  or  is  not  equal  to  zero.  Suppose  first  ^'3,  not 
zero.     We  know  now  that  each  of  the  quantities 


^„, 

^., 

'^IS* 

^., 

K. 

^23. 

^3,, 

<^32. 

^33 

is  zero,  and  consequently  that  the  substitution  'JT,  which  is  any  sub- 
stitution whatever  of  the  group  (3,  gives  as  the  transforms  of  ^j ,  j/^ » 
/g  linear  functions  of  j', ,  .  .  .  ,  jj/^  only.  Denote  by  XT?  XT'  •  •  •  the 
different  substitutions  of  (3  ;  by  F,  F',  .  .  .  the  corresponding  trans- 
forms of  y^ .  Each  of  these  functions  depends  only  on  J', ,  jKa ,  jKg ,  J4 , 
and  consequently  the  function-group  formed  by  the  linear  functions 

aY^a'Y'  -^  .  .  . 

can  contain  at  the  most  but  4  distinct  functions.  Let  F, ,  .  .  .  ,  F^ 
denote  these  functions,  where  k  ^  4. 

Further,  we  remark  that  substitution  in  (5  changes  the  functions 
of  the  series  F,  F',  .  .  .  among  themselves.  Suppose  XT,  changes  F 
into  F, ;  we  know  that  XT  changes  y\  into  F,  and  consequently  'Q^'G;, 
changes  jj  in  Fj ;  and  since  tTITi  is  a  substitution  of  the  group  (3,  it 
follows  that  F,  belongs  to  the  series  F,  F',  .  .  .  .  The  same  reason- 
ing of  course  applies  to  the  general  case,  and  in  this  case  we  see 
that  the  substitutions  of  (3  transform  into  one  another  the  functions 
of  the  function-group  orF-j- «;'F' -J"  ....  The  number  of  distinct 
functions  Fj ,  .  .  .  ,  F^  contained  in  this  function-group  being  less 
than  the  number  of  variables,  it  follows  that  (3  cannot  be  prime. 
The  substitutions  of  (3  can  obviously  be  put  in  the  form 


(61) 


F,    ...  F,;    a^\-\-  ...  + «,F,,  ...,/?,  F, +  ...+ A  ^ 
n  +  ,...     ;    ^,F,+  ...  +  ;.,n-f  ^,  +  ,F,  +  ,+  ...,      ... 


APPLICATION   OF    THE    THEORY  OF  SUBSTITUTIONS.       44I 

In  the  second  case,  suppose  b\^  =  o,  and  let  V,  V,  .  .  .  be  the  trans- 
forms of  J3  by  the  substitutions  of  (5-  None  of  these  can  contain  ^3 , 
and  consequently  the  function-group  formed  from  V,  Y',  .  .  .  must 
contain  a  number  of  distinct  functions  V^  ,  .  .  .  ,  V^.  less  in  number 
than  the  variables  Ji ,  ^'j ,  /s ,  J\ ,  ^, ,  ^„ ,  -i^j .  The  same  sort  of  reason- 
ing as  above  will  conduct  us  to  the  form  (61)  of  the  substitutions  in  (3. 
We  will  suppose  now  that  the  substitutions  of  (3  have  all  been 
brought  to  the  form  (61).  Consider  the  characteristic  equation  cor- 
responding to  a  substitution  of  (3  of  this  form  ;  the  first  member  of 
this  equation  is  evidently  divisible  by  the  first  member  of  the  char- 
acteristic equation  corresponding  to  the  substitution 

{62)     |F.  ...  F,;  ^,F.-f  ...-f«',n,    ...,    A5^.  +  .-.+Ani. 

But  by  hypothesis  all  the  substitutions  of  (3  have  for  the  right- 
hand  side  of  their  characteristic  equations  a  power  of  .f  —  i  ;  it  fol- 
lows then  that  the  right-hand  side  of  the  equation  corresponding 
to  (62)  must  also  be  a  power  of  ^  —  i.  The  group  (3^  formed  by  the 
partial  substitutions  (62)  cannot  therefore  be  prime  if  ;^  >  i,  and  so, 
by  a  proper  choice  of  the  independent  variables,  the  substitutions  of 
this  group  can  be  put  in  the  form 


(63) 


¥,...¥,     ;       0.(F,  .  .  .  F,).  .  .  .  ,     0,(F,.  .  .  F,) 
F,+,  .  .  .  F,;  0,+,(F,  ...  F,),  ...  ,     cp„iV,  .  .  .  F,) 


where  /  <  /&  and  (p  denotes  a  linear  function   of  the  quantities  in 
parenthesis. 

If  />  I,  we  can  again  so  choose  the  variables  V^  .  .  .  Vi  that  the 
partial  substitutions 

(64)  I  F,  .  .  .  F,;     0.(F.  ...¥,),     .  .  .  ,     0.(F.  .  .  .  F,)  | 

shall  take  the  simpler  form 

F, . . .  F„,  ;  t,{v,. . .  F„,), . . .,  My,-  ■  •  y,n) 

F,„+,  .  .  .  F,;  ^v+.(F,  .  .  .  F,),    .  .  .  ,  UV,  .  .  .  F,) 


(65) 


where  vi  <  /.  By  continuing  this  process  we  can  obviously  so  choose 
the  independent  variables  F  that  the  first  one,  F, ,  shall,  for  each 
substitution  in  (3,  be  changed  into  itself  multiplied  by  a  constant : 


442 


LINEAR  DIFFERENTIAL  EQUATIONS. 


and  since  the  characteristic  equation  corresponding  to  each  substi- 
tution in  (5  is  a  power  of  ^  —  i,  it  is  obvious  that  this  factor  reduces, 
to  unity.     We  have  then  the  theorem  : 

If  each  substitution  in  0  has  {s  —  i)"  =  O  for  its  characteristic 
equation,  there  must  exist  at  least  one  function  of  the  variables  y^  .  .  .  j'„. 
which  is  not  altered  by  mty  substitution. 

There  may  obviously  be  more  than  one  such  function.  Suppose 
Y^  .  .  .  Yp  are/  distinct  functions  possessing  this  property ;  then  the 
substitutions  of  (3  can  be  put  in  the  form 


(66) 


Y. 
F. 


K 


F, 


F. 


p+i 


.F„;  ^,  F,  +  . . .  +  ««F„,  . .  . ,  AF,  +  . . .  +  y5„F„ 


Now  the  first  member  of  the  characteristic  equation  of  such  a  sub- 
stitution is  equal  to  the  product  of  {s  —  i)^  by  the  first  member  of 
the  characteristic  equation  belonging  to 

(67)1   F,+,  ...F„;  «,+,F,+,...  +  «„F„,...,/?,+,F^+,...+/?„F„|  .. 

Substitutions  of  this  form  couresponding  to  the  different  substitu- 
tions of  (3  will  then  have  {s  —  i)"~^  =  o  for  their  characteristic  equa- 
tion. We  conclude,  then,  that  we  may  so  choose  the  variables  as  tO' 
make  these  substitutions  of  the  form 


(68) 


F 


/•+1 


F 


P+g+T- 


Y„  ;  a  V:  F,+, .  .  .  +  a''„  F,  . . .  /3'p+,  F,+,  +  . .  .4-/?'„  F„ 


By  a  continuation  of  this  process  we  see  that  all  of  the  substitu- 
tions of  (3  can  be  put  in  the  form 


(69) 


Y.  .  .  Y, 


F 


K 


F 


p+i 


F 


q+l    • 


Yq ;  F^_,_, 4-/^+1 , 

Yr\       F^_,_,   -[-/j  +  , 


Yq  +  fq 

Y,.+fr 


where /^+,.  .  ./^  are  linear  functions  of  F^,.  .  .  ,Yp\  fj^^  .  .  ./^are 
linear  functions  of  F, ,  .  .  .  ,  F^;  etc. 

As  every  substitution  in  (3  has  now  this  form,  the  elementary 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       443, 

substitutions  H,  3B,  •  .  .  ,  from  which  (3  is  derived  must  also  have 
the  form  of  (69). 

We  arrive  now  at  the  second  part  of  our  problem.  So  far  we 
have  assumed  that  all  the  substitutions  in  (5  have  unity  as  the 
single  root  of  all  their  characteristic  equations,  and  on  this  hypo- 
thesis have  found  the  form  (69)  for  each  of  the  substitutions.  We 
have  now  to  show  how,  for  any  group  (3,  we  can  ascertain  whether 
or  not  the  substitutions  H,  !fiS,  .  .  .  are  of  this  form.  To  do  this 
we  will  endeavor  by  the  method  of  indeterminate  coefficients  to 
find  out  whether  or  not  there  exist  linear  functions  of  the  inde- 
pendent variables  )\,  .  .  .  , y,^  which  are  unaltered  by  the  substitu- 
tions H,  !fiS  ,  .  .  .  We  will  assume  certain  linear  functions  of  j^'j ,  .  .  .  ,  f„ 
of  the  form,  say, 

o^.y^  +  •  •  •  +  ^«yn, 

and  operate  upon  them  by  the  substitutions  H,  !B,  .  .  .  If  the  trans- 
forms of  such  functions  are  the  same  as  the  functions  themselves, 
we  will  arrive  at  a  series  of  linear  homogeneous  equations  for  the 
determination  of  the  constants  a^,  .  .  .  ,  a,,,  the  number  of  these  equa- 
tions being,  of  course,  in  general  greater  than  Ji.  It  may  happen  that 
these  equations  can  all  be  satisfied  (as  certain  determinants  may 
vanish)  and  still  leave  a  number,  say  /,  of  the  coefficients  a  arbi- 
trary. If,  however,  we  have  /  arbitrary  constants,  we  will  also  have 
/>  functions  Fj,  .  .  .  ,  V^  possessing  the  required  property;  viz.,  the 
transforms  of  Fj ,  .  .  .  ,  F^  by  each  substitution  in  (5  will  be  again 
Fj ,  .  .  .  ,  F^ .  The  next  step  is  to  ascertain  whether  or  not  there  exist 
functions  Vf,+,,  .  .  .  ,  F^  whose  transforms  by  the  substitutions  of  (5 
are  equal  to  themselves  increased  by  linear  functions  of  F, ,  .  .  .  ,  Vp. 
We  thus  arrive  at  another  system  of  linear  homogeneous  equations 
of  the  first  degree,  etc.  If  in  continuing  this  process  we  never  arrive 
at  an  incompatible  system  of  linear  homogeneous  equations  we  may 
take  Fj ,  .  .  . ,  V^,  Vp^j, .  .  . ,  F^ ,  .  . .  for  the  system  of  variables  which 
shall  throw  the  substitutions  H,  35,  .  .  .  ,  and  consequently  all  the 
substitutions  of  (B,  in  the  form  (69).  The  transition  to  the  substitu- 
tions A,  =  rtH,  B  =  dJS,  ...  of  the  group  G  is  at  once  effected  by 
multiplying  the  transforms  in  (3  hy  a,  d,  .  .  .  Since  now  each  of  the 
substitutions  of  G  multiplies  each  of  the  functions  V^,  .  .  .  ,  I^  by  a 
constant  factor,  the  problem  proposed  concerning  this  group  is  solved.. 


444  LINEAR  DIFFERENTIAL  EQUATIONS. 

We  will  now  take  up  the  case  where,  in  following  the  above  in- 
dicated process,  we  arrive  at  a  system  of  incompatible  linear  equa- 
tions for  the  determination  of  the  coefficients  a\  if  this  case  presents 
itself,  we  know  at  once  that  <3  contains  at  least  one  substitution 
whose  characteristic  equation  has  several  distinct  roots.  To  solve 
the  problem  which  thus  presents  itself,  we  have  to  do  one  of  two 
things  ;  viz.,  we  may  either  determine  a  function-group  W  containing 
less  than  n  distinct  functions,  or  we  may  determine  a  substitution, 
say  5,  whose  characteristic  equation  contains  several  distinct  roots. 

We  will  assume  for  our  independent  variables  those  which  shall 
give  the  substitution  H  in  its  canonical  form,  and,  for  simplicity,  will 
take  the  special  case  above  considered,  where 


(70)         H  = 


^xi  "if  ^zi       '■)    -s"!  ~r  /i  >  ^2~T'  y^j  •s'3  ~r  y^ 


We  will  now  determine  the  transforms  of  }\  by  H,  !fi5,  .  .  .  ;  if 
among  these  transforms  there  are  any  which  are  linear  functions  of 
the  others  and  of  j;/, ,  we  will  discard  them  ;  suppose  the  remaining 
distinct  functions  to  bej)-'/,  .  .  .  ,  j,''.  Operate  now  on  j//,  .  .  .  ,;)','' 
by  H,  !fiS,  .  .  .  and  obtain  new  functions,  from  which  discard  again  all 
those  that  are  linear  functions  of  the  others  and  of  j, ,  j/,  .  .  .  ,  j?/,''; 
if,  again,  there  remain  new  independent  functions  J,*"^',  •  •  •  ,  y^  , 
we  will  transform  them  by  H,  IB,  .  .  .  and  discard  as  before.  By 
continuing  this  process,  which  is  necessarily  limited,  since  the  total 
number  of  distinct  functions  cannot  exceed  the  number  of  the  in- 
dependent variables  jj/, ,  JF2 ,  JJ'3 ,  JF4 ,  we  shall  finally  come  to  a  point 
where  every  new  function  obtained  by  the  operation  of  the  sub- 
stitutions H,  JS,  .  .  .  is  a  linear  function  of  the  preceding  ones. 
The  linearly  distinct  functions  so  obtained,  say  j', ,_;//,  .  .  .  are  evi- 
dently changed  into  linear  functions  of  themselves  by  H,  ^,  .  .  . 
(and  consequently  by  all  the  substitutions  of  (B).  The  problem  now 
divides  into  the  two  cases  above  mehtioned.  First :  Suppose  that 
none  of  the  functions  j\  ,  j\',  .  .  .  contain  any  of  the  variables 
s^ ,  s„,  s^;  then  the  number  of  these  functions  is  at  most  equal  to  the 
number  of  the  functions  j/^ ,  y„ ,  y^ ,  j,, .  We  can  at  once  proceed  to 
the  general  case  of  «  independent  variables  _;/,  and  so  see  at  once  that 
we  can  obtain  a  system  of  less  than  71  functions  such  that  the  sub- 


APPLICATION  OF    THE    THEORY  OF  SUBSTITUTIONS.       445 

stitutions  of  Q>,  and  consequently  those  of  G,  transform  these  func- 
tions into  Hnear  functions  of  themselves  without  introducing  any 
new  functions.  The  function-group  W  formed  from  y^ ,  j/,  .  .  . 
gives  us  the  solution  of  our  problem. 

Secondly  :  Suppose  that  among  the  functions  j\  ,  _y/,  .  .  .  there 
is  one,  say j>/,p,  which  contains  s.^,  s^,  s^;  we  will  then  stop  the  pre- 
ceding series  of  operations  when  we  arrive  at  j/^p  ,  and  by  retracing 
our  steps  find  the  substitution,  say  TI,  which  changes  j/^  into  j'/ .  This 
substitution  will  of  course  be  of  the  form  (39),  and  the  coef^cients 
<^,j ,  .  .  .  ,  ^^33  may  or  may  not  satisfy  equations  (43),  (44),  and  (45). 
Suppose  the  coefificients  d^j  do  not  satisfy  these  equations ;  we  can 
then  determine  a  substitution  H^XC  whose  characteristic  equation  shall, 
have  several  distinct  roots.  The  first  member  of  this  equation  will 
be  the  same  as  the  first  member  of  (41),  and  on  being  developed  will 
be  of  the  form 

s"  +  (ps"-'  -\-  (p^s"-^  -f  .  .  .   =  o. 

It  is  only  necessary  here  to  assign  a  value  to  X  which  shall  not  satisfy 
equations  (37).  This  can  be  done  by  a  number  of  trials,  at  most 
equal  to  nr,  where  r  is  the  degree  in  A  of  one  of  the  coef^cients 
0,  0,  ,   .   .  . 

If  equations  (43),  (44),  and  (45)  are  satisfied,  we  can  choose  the  in- 
dependent variables  so  that  equations  (50)  shall  also  be  satisfied.  If, 
further,  we  have  i>^^d^^  =  o,  we  can,  as  shown  above,  so  choose  the 
variables  that  both  d^^  and  d„^  shall  be  zero,  but  d^^  shall  not  be  zero. 
We  now  proceed  to  determine  the  transforms  of  j/^  by  the  substitu- 
tions H,  !fiS,  .  .  .  ,  neglecting  all  transforms  which  are  linear  functions 
of  the  remaining  ones  and  of  y^ .  Repeat  this  process  on  the  new 
functions  so  obtained,  and  continue  in  the  manner  already  several 
times  described.  We  will  finally  arrive  at  a,  necessarily  limited, 
series  of  functions  jg ,  j/j',  .  .  .  ,  which  the  transformations  of  G  will 
transform  into  linear  functions  of  themselves.  Suppose  none  of  the 
functions  jj/3 ,  y/,  .  .  .  contain  ^, ;  then  their  number  is  at  most 
=  n  —  I,  this  being  the  number  of  independent  variables  when  s^  is 
omitted.  We  will  thus  have  a  system  of  less  than  n  functions  form- 
ing a  function-group  and  such  that  the  transformations  of  G  simply 
interchange  these  functions  among  themselves. 

Suppose  now  that  one  of  the  functions,  say  y/,  contains  ^, ,  and 


446 


LINEAR  DIFFERENTIAL  EQUATIONS. 


let  "CI  be  the  transformation  which  changes  y^  into  y^  ;  then  IH  will 
be  of  the  form  (50),  and  obviously  the  coefficient  b\^  will  not  be  zero. 
Since  neither  ^,3  nor  b' .^^  are  zero,  it  follows  that  equations  (5  5)  and  (57) 
cannot  be  simultaneously  satisfied.  If  IH  does  not  satisfy  (55),  then 
ID  =  XrH'^XH  will  be  of  the  form  (53),  and  we  can  determine  yu  so  that 
(54)  shall  not  be  satisfied,  and  then  determine  A  so  that  the  substitu- 
tion H'^'lt)  shall  have  a  characteristic  equation  other  than  {s  —  i)"  =  o. 
If  XH  does  not  satisfy  (57),  we  will  determine  //  in  such  a  way  that 
(56),  which  is  analogous  to  (55),  shall  not  be  satisfied,  and  then 
reason  with  ID  as  we  have  with  XH  in  the  previous  case. 

In  what  precedes  we  have  shown  how  to  determine  whether  or 
not  the  group  G  is  prime,  and,  if  it  is  not,  how  we  can  choose  its 
variables  so  as  to  throw  its  substitutions  in  the  form 


(71) 


F.  .  .  .    F,      ;    <t>iY, 


f;.), 


K 


If  the  group 
(72)      I  F.  . 


F„;   0.+.(F, 


F.;     cPiY, 


F„), 


,    0.(F.  . 


F.) 


f;.), 


0.(F,  .  .  .   F,) 


formed  by  these  partial  substitutions  is  not  prime,  we  can   again   so 
choose  its  variables  as  to  throw  its  substitutions  in  the  form  I  <  k: 


(73) 


Y,...Y,     ;   UY,  .  .  .    F,),         .  .  .  ,    /MF,  .  .  .    F,) 
F,+,  .  .  .    F,;    ,/',+, (F.  .  .  .    F,),    .  .  .  ,    ^MF,  .  .  .    F,) 


We  can  continue  this  process  and  finally  get  for  the  form  of  the 
substitutions  in  G 


(74) 


F,  ...  f;„    ;^^f,  +  ...  +  «„,f„„.  ..,  af,  +  ...+/?„,f„ 

y,n  +  .  .  .  .  F„ ;  K.F.  +  .  .  .  +  ;/„F„,  .  .  .  ,  cJ.F,  +  .  .  .  +  d,X. 
the  group  formed  by  the  partial  substitutions 
(75)   I  F.  ...  F,„;  ^,F,  +  ...  +  ..,„F„,,  ...,  ^^F,  +  .  .  .  +  A«F,« 
being  prime.     If  now  the  group  formed  by  the  substitutions 


(7^) 


^  m-\-i     •     •     '     •*  H  )       ^  m  -{- 1  ^  711  + 1  ~J~     •     '     •     n~  Vn  ^  n  > 
d„,  +  ,F,„  +  ,+     •     •     •     +    S„Yn 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       44/ 


IS  not  prime,  we  can  choose  the  variables  Y„^^^,  .  .  .  ,  F„  in  such  a 
-way  as  to  throw  these  substitutions  in  the  form 


in) 


■^  >« -|- I     •    •    •     ^  tn'  1        ^  »« + 1  -^  HJ  +  I     I      • 

P  m-\-\  ^  m-\-\      I       • 

F„,+,  .  .  .    F„    ;     /,„  +  ,F,„  +  .  +  . 
fV        V        4- 

'-'     JK  +  I   •*    Hi  +  I  I  • 


the  partial  substitutions 

<78) 


^ m-\-\   •     '    •     ■';«'>        ^  m  +  i  ^ m  +  i      |~ 
P  wi  +  i  -^  m-\- 1      I 


4-  or'   '  F  / 


-4-  /S'  /F  / 


forming  a  prime  group.     By  continuing  this  process  we  can  finally 
throw  all  the  substitutions  of  G  in  the  form 

F.        F        •         a  Y  A-         -A-  a   Y 
AF,+  ...+/5,„F,„ 

+  a',„.  Y,„  +  /„+.(F.  .  .  .  F,„),    .  .  .  ,     /?',„+,  F,„+,  +  .  , 
(79)       +/5',«'F„,+/„,(F,  ...  FJ 

■»  »/'+!   •   •   •     ^  }n"  »        '^    ;«'-|-i  ^  m'-\-i  ~t~   •   •   • 

+  «'',„.  F,„. +/„.,+,(  F.  .  .  .  Y„,),    ...,   yS'W,F„,,+,+  .  . 

~l     P     m"  ^  tit"  ~\    Jm"\^  \   '   •    '    ^  tn') 


the  partial  substitutions 


(So) 
(8i) 


m-\-  I 


-t  ,«  )  '^^i  -'^i      1~    •     •     •    "IT      ^m  ^  m  1         «     .     • 

AF,+  .  .  .+    /?,„F,„ 


forming  prime  groups. 


448  LINEAR  DIFFERENTIAL   EQUATIONS. 

We  can  now  show  how  to  determine  all  possible  function-groups 
which  are  such  that  the  substitutions  of  G  simply  change  among 
themselves  the  functions  of  a  given  function-group.  Denote  by  F 
such  a  function-group.  We  will  say  that  F  is  of  the  first  class  if  the 
functions  of  which  it  is  composed  depend  only  on  the  variables 
F, ,  .  .  .  ,  F,„ ;  of  the  second  class  if  some  of  its  functions  depend  on 
the  variables  Y„,j^^ ,  .  •  •  ,  Y,„>  without  containing  any  of  the  following 
variables,  etc.  Since  the  group  formed  by  the  substitutions  (80)  is 
prime,  there  can  exist  but  one  function-group  of  the  first  class  which 
will  be  formed  by  all  the  linear  combinations  of  F^  .  .  .  F,„ . 

We  will  now  give  the  means  of  determining  the  function-groups 
of  class  k  when  we  know  all  those  of  classes  inferior  to  k.  Knowing 
the  function-group  of  the  first  class,  we  can  then  proceed  to  build  up 
all  of  the  function-groups  of  higher  classes.  To  fix  the  ideas,  sup- 
pose /&  =r  3,  and  denote  by  F,  F' ,  .  .  .  the  different  function-groups 
of  the  first  and  second  classes,  and  by  ^  one  of  the  unknown  func- 
tion-groups of  the  third  class.  Denote  by  P,  P^ ,  .  .  .  linear  functions 
of  F„,'_|.i ,  .  .  .  ,  V,„"  ',  hy  Q,  Q^  .  .  .  ,  linear  functions  of  Fj ,  .  .  .  ,  F„,/; 
then  the  different  functions  of  ^  will  be  of  the  formP-j-  Q,  P^A^  Q^, 
.  .  .  Now  the  substitutions  of  G  (^dien  G  is  in  the  form  (79))  trans- 
form among  themselves  the  functions  P -{- Q,  P^-\- Q^,  .  .  .  .  In 
order  that  this  may  be  so  it  is  evidently  necessary  that  the  substi- 
tutions 


(82) 


F  / 1  F  //  •     a"  /  i    F  '  I    -4-  -4-  a"  n  Y  // 

•     •     *     )        P     ;«'-(-i  ^  m'-\-i      I"    •     •     •    "I      P     til"  ^  i> 


transform  among  themselves  the  partial  functions  P,  P^,  .  .  .  .  But 
by  hypothesis  these  substitutions  form  a  prime  group,  say  F,  and 
consequently  the  function-group  formed  by  P,  P, ,  .  .  .  will  contain 
all  the  linear  functions  of  F,„'_|_i ,  .  .  .  ,  Yj„» ,  and  in  particular  will 
contain  Y,„'^^ .  It  follows  then  that  ^  will  contain  a  function  of 
the  form 

(83)         /  =  F„,,+,  +  G  =  Y.n'+.  +  A,  F,  +  .  .  .  +  \,„  Y„, . 

Form  now  the  transforms  /',  f",  ...  of  this  function  by  the  substi- 
tutions A,  B,  .  .  .  ,  considering  Aj ,  .  .  .  ,  A„„/  as  indeterminates,  and 
discard  all  transforms  which  are  linear  functions  of  the  remaining 


I 


APPLICATION  OF   THE    THEORY  OF  SUBSTITUTIONS.       449 

ones  and  of/".  We  will  continue  this  process  in  the  manner  which 
has  already  been  several  times  described,  and  finally  arrive  at  a 
series  of  independent  functions  f,  f,  .  .  .  ,  f* .  That  the  series  of 
operations  which  give  rise  to  new  functions  which  are  linearly  inde- 
pendent of  the  preceding  ones  is  limited  is  easy  to  see  ;  for  the 
transforms  which  are  obtained  by  the  successive  substitutions  are 
linear  functions  of  F,„'+i ,  .  .  .  ,  Y„,"  and  the  in'''  products  of  A,  Fj , 
.  .  .  ,  A,„'  Y,„'  .  The  number  of  the  functions  f,  f,  .  .  .  ,  /'  is  thus 
at  most  =  iu"  —  ;//  +  ^^^"-  ^Y  combining/",  /',  ...,/'  linearly  we 
shall  obtain  a  function-group  whose  functions  are  transformed  the 
one  into  the  other  by  all  the  substitutions  of  G.  These  functions 
have  the  forms  P -{- Q,  P'  -\-  Q',  .  .  .  ,  P' -\-  Q\  where  P,  P\  .  .  .  ,  P' 
are  linear  functions  of  F,„'-fi ,  .  .  .  ,  Vm",  and  Q,  Q',  .  .  ■  ,  Q  are  similar 
functions  of  F, ,  .  .  .  ,  ¥,„•.  The  substitutions  of  T"' permute  among 
themselves  the  functions  of  the  function-group  formed  by  the  linear 
combinations  of /*, /",  .  .  .  ,  P* ;  but /^  is  prime  by  hypothesis.  There- 
fore, among  the  functions  P,  P' ,  .  .  .  ,  P^  which  are  formed  by  aid  of 
the  m"  —  m'  variables  F,„'+i,  .  .  .  ,  Y„,» ,  there  will  be  in"  —  in' 
distinct  functions. 

Suppose  that  P,  P' ,  .  .  .  ,  P"'"-'"'—^  are  these  distinct  functions, 
and  let  0  =  |P  -|-  CS^  be  any  one  of  the  functions  of  the  function- 
group  ^.  The  function  p  of  the  variables  F,„'_|_i ,  .  .  .  ,  ¥„,»  can  be 
written    as   a   linear   function    of   the   in"  —  in'  distinct   functions 

P     p'  pm"-m'-i 

■'>-'>•••>-'  • 

For  example,  let 

p  =  dP-{-  .  .  .  +  d'""-'"'-^P"'" -»''-'', 

then  evidently 

<{)  =  df  -\-  .  .   .  -|-^'«"-'«'-i/'«"-«''-'-|-Qj, 

where  (^^  depends  only  on  the  variables  F, ,  .  .  .  ,  Y„,, .  The  function- 
group  ^  will  now  be  obtained  by  combining  the  in"  —  in'  functions 
P,  .  .  .  ,  P'""-"''-!  with  the  functions  (^,,  <^^,  .  .  .  which  depend 
only  on  the  variables  F, ,  .  .  .  ,  ¥„,, .  It  may  happen  that  all  the 
functions  (S^i ,  (Ji^ ,  .  .  .  reduce  to  zero  ;  but  if  they  do  not,  they  will 
form  a  function-group  whose  elements  are  permuted  among  them- 
selves by  the  substitutions  of  G.  The  function-group  so  formed  will 
then  be  one  of  the  assumed  function-groups  i% /^',  .  .  .     Among  the 


450  LINEAR  DIFFERENTIAL   EQUATIONS. 

functions  CJi ,  di^ ,  .  .  .  it  is  easy  to  determine  those  which  belong  to 
the  function-group  derived  from  /,  /',  .  .  .  ,  /' ,  this  function-group 
being  obviously  a  minor  of  ^.  In  order  to  obtain  the  required 
functions,  we  remark  that  by  subtracting  properly  chosen  linear  func- 
tions of  /,  /',...  ^f»'"-m'-z  fi-on^  jrm"-m'^  _  ^  .  , /'  wc  obtain  func- 
tions CSi '""-'"',  .  .  .  ,  CS^^  which  contain  only  the  variables  Fj ,  .  .  .  ,  Y^i\ 
and  further,  the  coefficients  of  the  variables  in  these  functions  will 
be  linear  in  A^ ,  .  .  .  ,  /l,„' .  If  we  wish  to  have  d^, ,  Q, ,  .  .  . 
reduce  to  zero,  it  will  be  a  fortiori  necessary  that  the  functions 
^m"-m' ^  .  ,  .  ,  iS^'^  which  form  a  part  of  the  series  CSi, ,  (Si^'  •  •  •  shall 
vanish.  In  order  that  this  may  happen,  it  is  obviously  necessary  that 
all  of  the  coefficients  of  Fj ,  .  .  .  ,  F^^/  shall  vanish  in  each  of  the 
functions  Q,'«"-'«',  .  .  .  ,  CSi' ;  we  will  then  have  for  A, ,  .  .  .  ,  \^>  a 
system  of  linear  equations  in  general  greater  in  number  than  m' . 
If  these  equations  are  not  incompatible,  then  to  each  system  of 
values  of  the  A.'s  which  satisfy  them  there  will  correspond  a  function- 
group  ^  derived  from  the  functions/", /"',  ,  .  .  ^f»'"-f"'-^. 

If  we  require  the  functions  (2^1,(2^2,.  .  .to  form  by  their  linear 
combinations  a  function-group  included  among  those  already 
known,  «>.,  F,  F' ,  .  .  .  ,  it  is  necessary  that  (2^»^"-w'-i^  ,  ,  .  ,  Qf  shall 
belong  to  this  function-group,  say  F.  Now  let  x,  x'f  -  •  •  be  the  dis- 
tinct functions  in  F;  all  the  functions  of  F  will  then  be  of  the  form 

FX  +  m'x'  +  •  •  •  , 
and  we  must  have 

<Sl'«"-"''+'  =  /M +  ////+.  . ., 

Replacing  in  these  equations  (^>""->"\  .  .  .  ,  (2^%  j,  j',  .  .  .  by 
their  values  in  K, ,  .  .  .  ,  Y„,'  and  equating  to  zero  the  coefficients  of 
each  variable,  we  shall  have  a  system  of  linear  equations  for  the 
determination  of  Aj ,  .  .  .  ,  A,„/,  /<,  //,  .  .  .,//,,  ///,  ,..,....  If 
these  equations  are  incompatible,  there  will  be  no  function-group  0  of 
the  third  class  containing  F;  if  the  equations  are  compatible,  then 
to  each  system  of  solutions  there  will  correspond  a  function-group  0. 
We  may  remark,  however,  that  if  In  the  expression 

/=  }v+,+A,F.+  .  .  .-f  A,„,F^, 


APPLICATION  OF   THE    THEORY  OF   SUBSTITUTIONS.       45 1 

^,,  .  .  .  ,  \,n>  have  had  values  given  them  which  permit  us  to  deter- 
mine a  function-group  ^  of  which  /^  is  a  minor,  we  can  obtain  an 
infinite  number  of  such  systems  of  values  of  these  constants  each  of 
which  will  give  rise  to  the  same  function-group  by  simply  consider- 
ing its  functions  in  the  form 

■where  v,  r',  v",  .  .  .  are  arbitrary  constants. 


CHAPTER   XII. 

equations  whose  general  integrals  are  rational, 
halphen's  equations. 

A  SPECIAL  class  of  Fuchs's  regular  equations,  that  is,  equations 
all  of  whose  integrals  are  regular,  is  the  class  of  equations  all  of 
whose  integrals  are  algebraic,  and  a  still  more  special  class  is  that  in 
which  the  general  integral  is  rational.  The  investigation  of  equa-^ 
tions  whose  general  integrals  are  algebraic  is  reserved  for  Vol.  II, 
but  a  brief  account  will  be  given  here  of  the  equation  whose  general 
integral  is  rational.     Write  the  equation  in  the  form 

(')        ^^')  =  g  +  /',^^  +  i^.^^+.  .  .  +  /'.,^  =  o, 

and  let  x\,  x^,  .  .  . ,  Xp  denote  its  finite  critical  points  ;  then,  since  the 
equation  is  to  be  in  Fuchs's  form,  we  have 

p         ^M 


where  Fk  is  a  polynomial  in  x  of  degree  k  (p  —  i)  at  most,  and 

ip{x)  =  {x-  x,){x  -X^)  .    .   .  {x  -  Xp). 

As  by  hypothesis  the  integrals  of  (i)  can  only  have  poles  as  critical 
points,  it  follows  that  the  roots  of  the  indicial  equation  for  each  of 
the  critical  points  x^,  x^,  .  .  .  ,  Xp  must  all  be  integers,  and,  further, 
that  no  logarithms  can  appear  in  the  expressions  for  the  integrals. 

For  any  given  equation  we  can  find  at  once  whether  the  first  of 
these  conditions  is  or  is  not  satisfied  by  simply  forming  the  indi- 
cial equation  corresponding  to  each  critical  point  and  obtaining  its. 
roots.  That  the  second  condition  may  also  be  satisfied  the  equation 
must  be  such  that  equations  (47)  of  Chapter  IV  are  satisfied  for  all 
the  groups  of  integrals  belonging  to  each  finite  singular  point. 

45  a 


EQUATIONS    WHOSE   GENERAL   INTEGRALS  ARE   RATIONAL.     453 

Supposing  all  these  conditions  fulfilled,  we  see  that  the  general 
integral  is  uniform  throughout  the  plane  since  its  singular  points 
at  a  finite  distance  are  all  poles.  To  show  that  it  is  not  only  uni- 
form but  also  rational,  it  is  only  necessary  to  show  that  the  point 
infinity  is  also  a  pole.  To  show  this  we  form  the  indicial  equation 
for  the  point  4:  =  co .  Divide  the  roots  of  this  indicial  equation  into 
groups,  the  roots  in  each  group  differing  from  each  other  only  by 
integers.  Suppose  the  smallest  root  in  each  group  to  be  denoted 
by  a,  a' ,  .  .  .  respectively,  and  the  number  of  roots  in  each  group 
to  be  denoted  by  /J,  ft' ,  .  .  .  respectively.  For  very  large  values  of 
-X  the  general  integral  will  be  of  the  form 

—     00+0.    log-+0,    log'-+.    .    •  +0a-x   log«-'-J 

+  ^  [00'  +  0/  log  I  +  0/  log^  1+'"+  <P'^'-  ^  log  «'-'  \\ 
+ , 

the  functions  0,  0',  0",  .  .  .  being  series  going  according  to  ascend- 
ing integral  powers  of  - .     As  already  seen,  however,  this  expression 

must  be  uniform,  and  so  the  logarithms  must  disappear  and  the  ex- 
ponents a,  a' ,  .  .  .  must  be  integers.  It  follows,  therefore,  that  the 
general  integral  has  the  point  infinity  as  a  pole,  and  so  is  a  rational 

P 
function  of  the  form  yc '  P  ^^^  Q  being  polynomials  in  x.     These 

polynomials  are  very  readily  found  ;  the  denominator  Q  is  known  at 
once,  since  from  the  differential  equation  and  the  various  indicial 
equations  we  know  the  finite  poles  of  the  general  integral  and  their 
respective  orders  of  multiplicity.  Suppose  the  poles  x^,  x^,  .  .  .  ,  x^ 
to  be  of  orders  of  multiplicity  a^,  a^,  .  .  . ,  ap  respectively;  then  ob- 
"viously 

Q  =  A{x  —  x^Yi(x  —  x^Y'  .  .  .  {x  —  XpYp, 

•where  y^  is  a  constant.  The  order  of  multiplicity  of  ,r  =  00  is  known 
from  the  development  according  to  powers  of—  ,  and  so  the  degree 


454  LINEAR  DIFFERENTIAL   EQUATIONS. 

of  the  numerator  P  is  known  ;  to  find  its  coefficients  we  have  sim- 

P  .  I 

ply  to  identify  the  development  of  -^  according  to  powers  of  -  with 

the  corresponding  development  furnished  by  the  differential  equa- 
tion itself.  In  connection  with  the  preceding  the  reader  is  referred 
to  a  note  by  Mittag-Leffler  in  the  Comptes  Rendus,  vol.  xc.  p.  218. 

Halphen's  Equations. 

A  rather  more  general  class  of  equations  than  the  preceding  has 
been  studied  by  Halphen.  Consider  those  equations  whose  inte- 
grals are  regular  and  uniform  in  every  region  of  the  plane  that  does 
not  contain  the  point  x\  .  The  other  critical  points,  x^,x^,  .  .  .  ,  Xp, 
must  now  be  merely  poles  of  the  integrals,  and  so  the  roots  of  the 
indicial  equations  corresponding  to  these  points  must  all  be  integers; 
and  further,  no  logarithms  can  appear  in  the  developed  forms  of  the 
corresponding  integrals.  Let  us  suppose  these  conditions  all  satisfied 
for  the  points  x^,  x^,  .  .  .  ,  Xp-,  we  can  now  find  the  general  integral. 
Consider  the  region  of  the  point  x^,  and  let  J„,  7, ,  •  .  •  ,  J*  denote 
one  of  the  groups  of  the  system  of  fundamental  integrals  belonging 
to  this  point.  If  r  denote  the  corresponding  root  of  the  indicia! 
equation,  we  have 

y,  =  {x-  X^Jl^  , 

Ji  =  {x  -  x^Yie.u,  +  u,i 

y,  =  {x-  xyid,i(,  -{-e,_,2(^-{- .  .  .4-  u,i 

where  n^,  u^,  .  .  .  ,  ?(/^  are  uniform  in  the  region  of  x^,  and 

27ti         '  •   •   •  '  f^*  -  ^j 

We  can  make  a  further  assertion  concerning  the  functions  //,  viz., 
they  are  rational  functions ;  in  fact,  the  points  x^,  x^,  .  .  .  ,  Xp  being 
ordinary  points  for  the  functions 

{x-x,y     and     \og{x-x,), 


HA  LP  HEN'S  EQUATIONS.  455 

and  poles  iox  y^,  y^,  .  .  .  ,  yk,  must  also  be  poles  for  ii^,  ?/, ,  .  .  .  ,  ti^\ 
these  functions  are  therefore  uniform  not  only  in  the  region  of  x^ , 
but  throughout  the  plane.     Again,  the  functions 


(—.)-•=?(■ -5,1  ■ 


and 


I  I  X  \ 

log  {x  —  X,),  =  -  log-  +  log  f  I  -  -M , 

are  regular  expressions  for  ;f  =  t>o  ;  the  same  is  true  for  _;Vo,Ji.  •  •  -.J*, 
and  consequently  for  7i^,n^,  .  .  .  ,  ii,..  These  two  properties  of  the 
functions  n  taken  together  show  that  they  are  rational  functions,  say 

?/„  —  ^  ,       U^  —   ^  ,      .    .    .  ,      «4  ^  . 

The  polynomials  P  and  Q  are  found  just  as  in  the  preceding  case ; 
we  know  the  poles  x^ ,  x^,  .  .  .  ,  Xp  and  their  respective  orders  of 
multiplicity,  and  so  the  denominator  Q  is  formed  at  once ;  a  develop- 
ment of  the  general  integral  for  very  great  values  of  x  will  give  us 
the  degrees  of  the  numerators P^,  P,,  .  .  .  ,  P,,;  to  obtain  their  coeffi- 
cients we  have  only  to  substitute  the  preceding  expressions  in  the 
differential  equation  and  identify  the  result  with  zero. 

Another  very  interesting  class  of  equations  also  due  to  Halphen  * 
is  the  class  where  the  general  integral  is  of  the  form 

y  =  c.e'^^-fix)  +  r,^"-''/,(A-)  +  .  .  .  +  c,r--xfXx\ 

where  f^,  f<^,  -  •  -  ,  fn  are  rational  fractions.  Halphen's  investiga- 
tion involves  certain  properties  of  differential  equations  whose  co- 
efficients are  doubly  periodic  functions  of  x.  The  following  inves- 
tigation is  due  to  Jordan: 

Consider  the  differential  equation 

d^'v  d"-'v  d"-^v 

*  Comptes  Rendus,  vol.  loi,  p.  123S. 


456 


LINEAR  DIFFERENTIAL  EQUATIONS. 


where  P^ ,  P, ,  .  .  .  ,  P„  are  polynomials  the  degree  of  any  one  of 
which  is  at  most  equal  to  the  degree  of  the  first  one,  P^ ;  this  is  of 
course  equivalent  to  saying  that  the  developments  of 


P.       P. 
P'     P' 


Pn 
P. 


according  to  decreasing  powers  of  x  contain  no  positive  powers  of 
X.     We  will  suppose  that  the  integral  of  (2)  contains  as  critical  points 
at  a  finite  distance  only  poles  ;  the  preceding  considerations  will  of 
course  enable  us  to  make  sure  of  this  fact  in  any  particular  case. 
Write  now 


(3) 


y  =  Rt, 


where  i?  is  a  rational  function  of  x.     Equation  (2)  now  takes  the 
form 


(4)    P.R%  +  nP.,R 
+    Pfi 


.t-t    ,   ni^jy 


dx"-' 


+  (;^-  I)    P,R' 
+  P.R 


d"-H 

dx"^^ 


=  0, 


and  this  is  of  the  same  form  as  (2) ;  for,  after  clearing  of  fractions, 

its  coefficients  will  be  rational  polynomials  in  x,  and  its  integrals 

possess,  by  (3),  only  polar  singularities;  finally,  if  we  admit  that  in 

the  development  of  R  according  to  descending  powers  of  x  the  first 

term  is  Ax^,  then  the  first  terms  in  the  developments  of  R',  R" ,  .  ,  , 

will  be,  to  a  constant  factor  pres,  x  to  the  powers  /  —  i,  p  —  2,  .  .  . 

respectively.     It  is  easy  to  see,  then,  that  after  dividing  by  P^R  the 

coefficients  of  (4)  can  contain  in  their  developments  no  positive  pow- 

d"-H 
ers  of  x\  for  example,  take  the  coefficient  of  -7—37,  viz., 


nP,R'  +  P,R 
P.R 


dx"- 

R'P. 
=  ''-r-^p: 


from  what  has  been  said  it  is  clear  that  no  positive  power  of  x  can 
appear  in  the  development  of  this. 

A  particular  form  of  the  preceding  transformation  will  now  be 
applied  to   equation  (2) ;   we  can  of  course  determine   in  advance 


HA  LP  HEN'S  EQUATIONS. 


4S7 


the  poles,  say  jr^,  x\,  .  .  .  ,  x^,,  oi  the  general  integral  of  this  equation, 
and  also  their  respective  degrees  of  multiplicity ;  say  these  are 
/^j ,  yUj ,  .  .  .  ,  yWp .     Now  transform  (2)  by  the  relation 


•^      {x  —  xy^ix  —  x.y-' 
the  result  of  the  transformation,  say 


{x  -  XpY, ' 


(5) 


(ft 


d'"'t 


Q.-JZJ.+ Q.ijz^.  +  '  •  •  +  Qnt  =  o, 


dx" 


dx 


is  of  the  same  type  as  (2),  but  its  integrals  have  no  poles.     Again, 
make 

(6)  t  =  e^^'v  ; 

the  new  transformed  equation,  viz., 

d^'-'v 


(7) 


or 
(8) 


+     Q. 


dx""- 


+  A"a 

+  A-^a 

+  ■ . . 

+  a 

V  —  0, 


d"z> 
dx'' 


^n  -r:n  +  ^1 


d'^- 


dx"- 


+  R,riJ  =  o, 


is  obviously  of  the  same  type  as  the  original ;  we  will  suppose  A  so 
determined  that  the  coefificient  of  the  term  of  highest  degree  in  R„ 
is  made  to  vanish  ;  R„  will  then  be  a  polynomial  of  lower  degree 
than  Rg . 

Denote  by  ^, ,  ^^,  .  .  .  the  roots  of  7?„  =  o ;  by  decomposition 
into  partial  fractions  we  have,  remembering  that  the  degree  of  R/,  is 
at  most  equal  to  the  degree  of  R^ , 


(9) 


f  =A+^ 


B, 


Xtix-^^'' 


where  A  and  B  are  constants,  and  in  particular  A„  =  o.     The  points 
4  being  ordinary  points,  in  the  region  of  which  the  integrals  are 


458  LINEAR  DIFFERENTIAL  EQUATIONS. 

regular,  the  index  /  can,  in  the  enumeration,  only  take  the  values- 

T      '>  k 

The  indicial  equation  relative  to  ^i  is 

(lo)     r{r  —  I)  .  .  .  (r  —  «  +  0  +  Bi^j{r  —  i)  .  .  .  {r  —  n  A^  2) 

+  Bi,,r{r  -  i)  .  .  .  (r  -  ;^  -f  3)  +  .  .  .  =  o  ; 

the  sum  of  the  roots  of  this  is 

/'TT^  n{n-\) 

(ii)  1 —  An- 

Now  since  B,i  is  an  ordinary  point  for  the  integrals,  these  roots  are 
necessarily  unequal  non-negative  integers;  the  least  values  which 
they  can  have,  therefore,  are  given  by  the  series  o,  i,  2,  .  .  .  ,  n  —  \  \ 
their  sum  is,  therefore,  at  least  equal  to 

nin  —  i) 

0+I+2+.    .    .+«_l:=-^^. 

It  follows  from  this  and  (ii)  that  ^/„  is  either  zero  or  a  negative  in- 
teger, and  a  fortiori  the  sum 

(12)  5=:s^,„ 

taken  over  all  the  points  ^  is  zero  or  a  negative  integer. 

dv 
Let  us  suppose  first  that  R^  is  not  zero,  and  take  v'  =  -y-  for  a 

new  variable.     The  equation  in  v  thus  becomes 
(-3)  ^.^^,  +  ^,^5^,+  ...+/?.'.  =  o; 

differentiating  this,  then  eliminating  v  between  (13)  and  the  nevv^ 
equation,  we  have 

^^4)  ^o^« S^  +  [(^0' + r:)R.  -  kr:^  -^  + . . . 


HALPHEN'S  EQUATIONS.  459 

This  is  obviously  of  the  same  type  as  the  equation  in  v\  the  ratio 
of  the  first  two  coefficients,  which  in  the  z^-equation  is  ^,  is  now 

R„      R„       R„ 


Now  let 


we  have  then 

r: 

R,       x-^. 

R^      X  —  V. 

R,  =  {x-  s:f\x  -  aT- 

R.  =  {x-v.t{x-V.r 


H h 


+-A 


the  sum  S'  formed  for  the  ^''-equation  in  the  same  way  that  5"  was 
formed  for  the  z'-equation  is  now  obviously 

(15)  S'  =  S-{-:Ea-  2ft. 

Since  2a,  the  degree  of  R^ ,  is  by  hypothesis  greater  than  2/3,  the 
degree  of  R„ ,  it  follows  that  S'  is  greater  than  vS.  In  like  manner, 
if  we  form  a  z'"-equation  in  the  same  way  as  we  formed  the  t/-equa- 
tion,  we  should  find  S"  >  S'.  If  we  could  continue  this  process 
indefinitely,  we  would  form  an  unlimited  series  of  increasing  integers 
S,  S',  S",  .  .  .  ,  none  of  which,  however,  are  positive,  which  is  absurd. 
It  must  be  then  that  there  is  an  equation  in  the  series  the  coefficient  of 
whose  last  term  is  zero.  Suppose  this  to  be  the  case  for  the  t''"-equa- 
tion;  this  equation  admits  a  constant  as  one  of  its  integrals,  and  so  the 
equation  in  v  admits  a  polynomial  7i:{x)  of  degree  7/1  as  an  integral, 
and  finally  the  /-equation  has  a  particular  integral  e^-^7t{x).    Make  now 

/  =  e^^nft^dx ; 

/,  satisfies  an  equation  of  order  n  —  i  which  is  of  the  same  type  as 
the  /-equation.  This  new  equation  therefore  admits  a  particular  in- 
tegral of  the  form  e^^'^nlx),  n^ix)  being  a  polynomial.     Again,  write 

t,  =  e^^^TtJt^dx, 


460  LINEAR   DIFFERENTIAL   EQUATIONS. 

and  continue  the  above  process.  We  of  course  come  at  last  to  an 
equation  of  the  first  order  whose  integral  is  of  the  form 

tn-i=  Cne^''--'^7Tn-i{x), 

where  c,^  is  an  arbitrary  constant.  The  general  value  of  /  is  now 
immediately  seen  ;  we  have  merely  to  retrace  our  steps  from  this  last 
equation,  performing  successively  the  indicated  integrations.  As  we 
know  how  to  effect  the  integrations,  we  can  see  immediately  what 
the  final  form  of  /  is ;  it  is,  viz., 

the  cs  being  arbitrary  constants  and  the  ^s  polynomials.  Now, 
since 

■^  ~  {x  —  xyi{x  —  x,)M2 .  .  .  (^-  —  xpytp' 

we  have  finally,  for  the  general  integral  of  the  equation  in  f, 

(16)  y  =  ^/«i-^/Xa-)  +  c,e^-<^xflx)  +  .  .  .  +  cnCOi«^flx\ 

f\ifi^'  •  '  ■>  fn  being  rational  fractions. 

Reciprocally,  every  differential  equation  whose  general  integral 
is  of  this  form  belongs  to  the  type  considered.  To  show  this  elimi- 
nate the  constants  c,,  between  (16)  and  its  derivatives,  and  suppress 
the  common  exponential  factors ;  we  will  thus  obtain  an  equation 
with  rational  coefficients  which,  by  clearing  of  fractions,  can  be  made 
integral.     Suppose  the  equation  is  then 

As  we  know,  its  integral  has,  at  a  finite  distance,  only  polar  singu- 
larities. It  remains  now  to  show  that  the  degrees  oi  P^,P^,  .  .  .  ,  P^ 
are  not  greater  than  that  of  P„ .  This  last,  however,  will  be  left  as 
an  exercise  for  the  student.  Halphen,  in  his  celebrated  "M^moire 
sur  la  n'duction  des  equations  diffcrcnticllcs  lineaires  aux  formes  in- 
tegrables,''  *  and  also  in  his  paper  in  the  Comptes  Rendus,  gives  the 

*  Savants  Etrangere,  t.  xxviii.  pp.  ill,  i8o,  273. 


HALF  HEN'S  EQUATIONS. 


461 


following  three  examples  of  this  class  of  equations,  viz.  {11  is  an 
integer  throughout) : 


(I) 


'dx^ 


'n{ii  -|-  i) 


+  « 


=  0, 


which  is  a  very  well  known  equation  ; 

^^^         dx^  ^    i^~  'dx  ~  \n^  "^ 


y  —  o, 


— in  this  11  must  be  prime  to  3  ; 

d^y       2n{ii -\^  \)  d'y      4/i{H-\-i)dy 


(III) 


dx*  x'        dx^  x'^        dx 

-n{ji  +  \){n  +  3)(«  -  2) 


+  ^ 


y  =  o. 


In  the  Comptes   Rendus  paper  the  following  examples  are  also 

given,  viz. : 

n\j\  "^'y      2{n+i)dy      (6n         \dy       2a 

this  has  one  solution  j/  =  ax''  —  2(271  —  i),  and  two  solutions  of  the 
form 

e^^'^-fix), 
where /is  rational. 


(V) 


d^y  2x      dy 

dx'^       x^  —  I  dx 


2a  n{ji  +  i) 

x"  -  i^         ~? 


+  (^  —  ^^(^  +  n  +  i)]  J  =  o  ; 
for  this,  supposing  n  positive,  there  are  two  solutions  of  the  form 


where /(.*■)  is  a  polynomial  of  degree  n  -\-  \. 

When  the  general  equation  is  restricted  to  have  the  single  critical 
point  ;ir  =  o  in  the  region  of  which  the  integrals  belong  to  the  ex- 


462 


LINEAR  DIFFERENTIAL   EQUATIONS. 


ponents  o,  i,  2,  .  .  .  ,  «  —  2,  ;?,  Halphen  shows  that  the   form  of 
the  equation  is 


+  (^„__^„.^._i^.)g_(^„_,.  +  ^.),. 


—     y—O,        ^ 


The  solutions  of  this  are  exponentials  t^,  where  rt:  is  a  root  of  the 
equation 

f{a)  =  a---\-A,a:'-'^  ,  ..+^„_,  =0; 

and  the  remaining  solution  is 

r    /wi 


CHAPTER   XIII. 

TRANSFORMATION    OF    A    LINEAR    DIFFERENTIAL    EQUATION. 
FORSYTH'S   CANONICAL   FORM.      ASSOCIATE   EQUATIONS. 

Although,  as  previously  stated,  it  is  not  intended  to  go  into 
the  theory  of  Invariants  in  the  present  volume,  it  is  nevertheless  de- 
sirable to  give  an  account  of  the  transformation  of  the  differential 
equation  and  its  reduction  to  the  canonical  form  adopted  by  Forsyth.* 

The  differential  equation  may  be  written  in  the  form 

^  ^"F  ^  d"-'V      n(u—i)      d"-'V 

This,  by  the  familiar  transformation 

and  subsequent  division  throughout  by  R^ ,  is  changed  into 

^'^      ^"+  2\n-2\     'dx^'^  3|  //  -  3|     '  dx^^  +  •  •  •  +  ^«.r  =  O. 

To  the  letters  P^,  P^,  .  .  .we  may  add  P„  and  P, ,  understanding, 
of  course,  that 

P,  =  1,     P,=  o. 

The  numerical  coefificients  are  obviously  equal  to 

n{n  —  i)       fi{H  —  \){n  —  2)       n{n  —  \){ii  —  2\n  —  3) 

—I  ^1  4| 

and  are  written  in  these  forms  by  Laguerre,  f  but  it  is  desirable  for 

*  Invariants,  Covariants,  and  Quotient-derivatives  associated  with  Linear  Differ- 
ential  Equations,  by  A.   R.    Forsyth.     Phil.   Trans,  of  the   Royal  Society,  vol.   179 

(1888),  pp.  377-489- 

f  Sur  quelques  invariants  des  equations  differeniielles  lineai^-es.     Comptes  Rendus, 
t.  88  (1879),  pp.  224-227. 

463 


464  LINEAR  DIFFERENTIAL   EQUATIONS. 

the  present  purpose  to  write  them  in  the  form  chosen  by  Forsyth. 
We  find  readily, 

p^  =  R,K  -  r:  -  {kr:  -  r:r^ 

p,  =  r:r,  -  3R0R.R.  +  2r:  -  {R^Rr  -  r:'r,), 


From  these  equations  we  see  that  the  functions  P  are  independent 
of  any  particular  choice  of  the  dependent  variable  j,  and  are  there- 
fore seminvariants  of  the  original  equation.     Suppose  now  that 

(i)  y  —  iiK 

where  A  is  a  function  of  x,  and  suppose  that  ii  satisfies  the  equation 

in  order  that  (i)  may  be  transformable  into  (ii),  z  must  be  some  func- 
tion of  X,  and  when  this  is  the  case  there  will  be  11  equations  con- 
necting A,  z,  X,  and  the  two  sets  of  coefificients  P  and  Q.  These 
equations  are  obtained  in  the  following  manner :  Making  the  sub- 
stitution y  =  tiX  in  (i),  we  have 

d"u  dl-      d^'-^Ji  ^1        d^\d"-^u 

^^^     ^"     "*"  ^  ^      dx"-'    '    2\n  —  2\'dx'  dx"-^  "r  •  •  • 

In  this  equation  we  have  to  change  the  independent  variable  from 
;ir  to  ^  in  order  to  compare  it  with  (ii).     Write 

(2)  2  =  (j){x) ; 

then,  by  a  formula  due  to  Schlomilch,  we  have 

dx'"-   ~ s=-i    s\    dz" 
where 

d"^ 

A^^,  =  Limit,    when  p  =  o,     of  -—^  \d){x-\-  p)—  (p{x) p. 


TRANSFORMATION :    FORSYTH'S  CANONICAL  FORM.        465 
Writing  in\  C^^^  =  ^««,s>  it  follows  from  this  last  that 

C.  =  coefificient  of  p"  in  f  p0'  +  -,  p=0"  +  -,p'(p'"  +...)'. 

Substituting  now  in  the  semi-transformed  equation  (a'),  the  coefifi- 
cient  of  -^—  is  readily  found  to  be 

dx  2   n  —  2  ^^ 

+  -7^=^ 1-7 As  s+  -^ r  PA  A^„.,  ,  -\-{n  -  2)—-A„.^, 

'    sl^n  —  sldx"-'       •     '    2|;^  — 2|     '[        «  -,5    1    v  j  ^^     „  3,, 

n  —  2\       d''-'-'X  ) 

4-  .  .  .  -1 ' A       V  4-  .  .  . 

'  ^   s\n  —  s—  2\  dx""-'     ^'3  j    ' 


t=o      {r\t\n  —  r  —  t\  dx' 

h        In  this  it  is  of  course  understood  that  P^^  i  and  /*,  =  O.     But  the 

present  form  of  the  equation  must  be  effectively  the  same  as  (ii),  and 

the  coefficients  of  corresponding  derivatives  of  7c  must  therefore  be 

proportional  to    one    another.      In    the    transformed    equation  the 

I  d'^u 
coefficient  of  -:  -7^  is  (here  s  =  n,  so  that  ji  =  o,  t  =  o  are  the  only 

values  for  terms  in  the   summation)  A„^„X,   or  the  coefficient    of 

d"u  .    A„„X  .     .    .    ,   , 

-7--  IS  — -. —  ;  that  IS,  it  is  Xz  ". 

dz"  n\ 

Hence  we  have 

n      y   'n      ^  ''  =  "-' '-"-^-'  \  n\  „    ^  da] 


s\n-s\^"  i!    r  =  o        t  =  o      \r_\t\n-r-tr  '■^''-''-'^''dx'  \ 


466  LINEAR  DIFFERENTIAL  EQUATIONS. 

or,  what  is  the  same  thing, 

n  —  s\  ,  r\  t\n  —  r  —  t\  dx* 


r  —  Q  i  —  o 


Write  now 


Wa=  -—A =^ P  - - 


+  .  .  .  +  /'.A, 


or,  symbohcally, 

(4)  PF«=(i,o,P,.  .  •P\^,  i)  A; 


then  the  coefficient  of 


■^n-r-t,s 

n  —  r  —  t\ 


in  the  foregoing  expression  is 


P,    d'X 


the  summation  extending  to  those  values  of  r  and  /  that  leave  the 
sum  r  -\-  t  unchanged  throughout — that  is,  the  coefficient  is 


and  therefore 


'^^QltZl.'n-'^^"  ^^IZ^^Yl 


n  —  s\ 


n  -6\ 


Changing  s  into  n  —  s  and  introducing  the  quantities  C  from  (3),  and 
this  becomes 


(iii) 


^^^  „'n  ^    /^  . . 


TRANSFORMATION :    FORSYTH'S  CANONICAL  FORM.         467 

If  it  were  desirable,  the  summation  in  this  might  extend  to  the 
value  B  ^=  n,  for  C„t,  m>  vanishes  if  in  <  m' .  Writing  (iii)  in  detail 
for  the  lowest  values  of  s  and  giving  P^  and  Q^  their  zero  values, 
we  have  in  succession 


1   -r'"  T  T 

.J  I      ti-s  —    ''  Q^n,n--i     I      ''  i^n-i,  «-3  "T"  ol         2^«-2,  «-3     |     VT 


;^  equations  in  all. 

There  is  a  Jirs^  invariant  of  the  differential  equation  which  is 
readily  derived  from  these  last  equations.  The  invariant  might 
properly  be  called  Brioschi's  invariant,  as  he  first  showed  that  it 
existed  in  the  case  of  differential  equations  of  the  third  and  fourth 
orders.  Forsyth's  deduction  of  this  invariant  (which  follows)  from 
equations  (5)^  (6)',  (7)',  involves  an  important  modification  of  the 
forms  of  these  equations.     We  have  from  (4) 

and  from  (3) 
while  generally 

^  m,  m  —  '^        >  • 

so  that  (5)'  is  now 

(5)  /V  +  i(«  -  i)A/' =  o. 

Writing 


468 


LINEAR  DIFFERENTIAL  EQUATIONS. 


we  have 

z"    =z'Z, 

z'"  =  z'{Z'  +  Z'), 

r  =  -  i{7i  -  \)\z, 

\"  =  -  !(;/  -  i)AjZ'  -  ^{n  -  i)Z\, 

r"  =  -  i{n -  i)x\z'  - 1(«  -  i)zz  + 1(71  -  \yz'\. 

Again, 

C«-.  =  ^V(«  -  2>"'-^J4^V-  +  ^{n  -  i)z"^\ 
:=  ^V(«  -  2)z'^-^\aZ  +  (3;^  -  5)Z=f, 

Introducing  these  and  the  values  of  A',  \"  in  (6)',  it  becomes 


(6) 


2Z  -Z'  = 


^^{p.-Q.n- 


The  values  of  C<,  «-3>    ^«-i,  «-3,    (^n-z.w-s  are  similarly  found  to  be 
C„,  „_3  =  Un  -  3).^'«-3]  2Z"  +  {4n  -  io)ZZ  +  (.e  -  2){n  -  3)Z=  [ , 

C_,,„_3  =  i(«-3>'«-3Z. 

By  means  of  these  and  the  above  values  of  A',  X",  X'",  (7)'  changes 
into 

(7)  Z"  -  3Z'Z  +  Z»  =  -^-  (P3  -  ^3^'  =)  -  -ip-  P.Z. 

Differentiating  (6)  with  respect  to  x,  and  remembering  that  Q^  is  a 
function  of  z,  and  that  ^■V  =  z"^Z,.  we  have 


Z"  -  zz  = 


6     /^/^,        ,,dQA         12 


2  ^/s'^ii-a 


;z  -|-  I  \  ^;ir  dz  j       n-{-  i 


2'  ^ZO  • 


TRANSFORMATION:    FORSYTH'S   CANONICAL  FORM.         469 
subtracting  (7)  from  this  gives 

71-];-  I  \  ax  dz  I       n-\-  \  ^ 

multiply  (6)  by  Z  and  subtract  from  this  last,  and  we  have 

6    m _ .- . f ^) --^ (A - y •&)  =  o, 

«  +  I  V  dx  dx  I       ;^  -f-  I  ^    '  ^^^ 

•or,  finally, 

(3f-.a.).'>  =  3f-./'., 

the  result  of  the  elimination  of  Z  between  (6)  and  (7).  This  last  re- 
sult is  Brioschi's  invariant. 

It  appears  from  the  preceding  investigation  that  there  exist 
rational  integral  functions  of  the  coefificients  of  a  linear  differential 
equation  and  their  derivatives  such  that,  when  the  same  function  is 
formed  for  the  transformed  equation,  the  two  functions  are  equal  to 
a  factor  pres,  which  factor  is  a  positive  integral  power  of  z' .  These 
functions  are  the  invariants  of  the  differential  equation,  and  the  ex- 
ponent of  z'  is  the  index  of  the  invariant. 

Reverting  now  to  equation  (5)  and  integrating  it,  we  have 

A^'*^'^-'^^  constant; 

since  equations  (iii)  are  homogeneous  in  the  dimensions  of  A,  this 
constant  may  have  any  arbitrary  value  other  than  zero  ;  taking  then 
the  value  unity,  we  have 


(iv)  X  =  z' 


/  -i(«-i) 


This  establishes  only  one  relation  between  the  quantities  z  and  A,  and 
we  may  suppose  z  arbitrary  so  far.     We  may  now  impose  any  other 


470  LINEAR  DIFFERENTIAL  EQUATIONS. 

condition  we  please  upon  z  and  A,  which  does  not  violate  (iv)  (or  (5)) 
say  such  a  condition  as  will  make  Q^  =  o.    By  (6)  we  must  then  have 


where,  by  (8), 


If  we  write 


12 

2Z'  —  Z''  A P 


z'  n  —  I  A 


the  equation  which  determines  Z  becomes 
d'^B  X 

and  so  we  may  write 

(10)  A=^«-',    z'  =  e-^. 

Hence  by  a  solution  of  a  linear  differential  equation  of  the  second 
order  we  can  remove  the  second  and  third  terms  of  the  general 
linear  differential  equation  of  any  order.  This  result  was  first  given 
by  Laguerre.*  The  modified  form  of  the  equation  to  which  we 
have  now  come  is  Forsyth's  canonical  forni^-\  and  for  the  future  we 
shall  speak  of  a  linear  differential  equation  of  order  n  as  being  in  its 
canonical  form  when  the  terms  involving  the  derivatives  of  order 
(«  —  i)  and  {11  —  2)  of  the  dependent  variable  are  lacking. 

Since  6  =  z'~^,  we  have  -7-^  =  f^'-V''  —  ^z'~^z'";  substituting 
in  (9),  we  get 

4  z'^       2z'^       »+^z'^ 

*Comptes  Rendus,  t.  88  (1879),  P-  226. 

+  Forsyth's  canonical  form  differs  very  little  from  Halphen's,  but  is  more  conven- 
ient in  computing  the  invariants. 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 

or,  on  reducing, 


^-•^^'  =v-l(?- 


\s,  x\  being  the  Schwarzian  derivative  -^ ( --j  . 


6P^ 


471 


Associate  Equations. 

In  Chapter  I,  equation  (42),  we  defined  Lagrange's  "  equation 
adj'ointe,''  and  there  spoke  of  it  as  the  "  adjunct  equation ;"  for  the 
future,  however,  we  shall  use  the  term  associate  equation.  *  The 
differential  equation  being  given  in  the  form 


(II) 


dx^'^^dx^-' 


its  Lagrangian  associate  is 

d"v        d"-"-  d"-^ 


.  -^R„j/  =  o, 


-{.(-iYvR„  =  o. 


Let  J^i,  Ji,  .  .  .  ,  x«  be  a  system  of  fundamental  integrals  of  (11); 
a  selection  of  any  n  —  i  of  them  will  suf^ce  to  determine  the  n  —  i 
coefificients  R.  Say  we  take  >'i,  Jj,  .  .  .  ,  Jn-i',  substituting  these 
in  (i  i)  and  solving,  we  have 


Ji 


^.= 


d"-y. 

d"-'-y^ 

dy. 

d»-'+y^ 

dx"-'      ' 

'  '  dx"-'--" 

dx"^ 

^^n-i+z. 

d"-y. 

d"-'-y^ 

dy. 

^n-t-\-zy^ 

dx"-'      ' 

'  '  dx"-'-"^ 

dx"" 

dx"-'+^ 

dn-2yn-.  ^n-i-^y^_^      dny^_^      ^«_/+,^^_^ 


dx""-' 


dx"-'-"^ 


dx"" 


^^n-i+i 


y. 


J^n-z 


-^^« 


*When  Chapter  I  was  written,  and  indeed  when  an  earlier  form  of  the  present 
chapter  was  written,  I  had  not  seen  Forsyth's  memoir,  and  had  not  been  able  to  find 
an  adopted  English  term  for  Lagrange's  ^'equation  adjointe"  so  I  used  the  word 
adjunct,  suggested  by  the  German  "  adjungirte,"  and  not  unlike  the  French  "adjointe.^' 
It  seems  better  now,  however,  to  employ  the  word  associate,  or,  when  speaking  sim- 
ply of  Lagrange's  "Equation  adjointe,"  the  word  adjoint.  It  is  unfortunately  too  late 
to  make  this  change  in  Chapter  I. 


472 


LINEAR  DIFFERENTIAL   EQUATIONS. 


where  the  subscript  n  in  z'„  indicates  that  the  function  jj/„  is  absent, 
and  where 


v^  = 


dx"-^         dx"-i 


dx"-^  dx"-^ 


d>^-y„.,     ^«-3j„_. 


dx"-' 


dx""-^ 


Substituting  these  values  in  (ii)  and  multiplying  through  by  z^„ , 
this  equation  becomes 


d"y 
dx^ 


d"-y 


dx""-^ 

dy^         d"-y. 


dx"" 

dy, 

dx"" 


dx"-^ 

d"-y^ 

dx"-^ 


dy 
dx 
dy 
dx 


y 

7i 


dy^ 

Tx     y^ 


dy„_,       d"-y„_^ 


dx"" 


dx""- 


I 
I 


=  o, 


or 


dx 


d"-y  d^'-y 

dx"-"" 

d"-y^ 


dx""-^ 

d'^-y^ 
dx"-^ 

d"-y, 

dx""-"- 


dx*"-^ 

d"-y, 

dx''-' 


d"-y„_,      d"~y„_. 


dx"-^ 


dx"- 


dy 
dx 

dy^ 
dx 

dy^ 
dx 

dy„-^ 
dx 


y 
y. 
y. 

yn-i 


=  o. 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 


473 


It  follows  therefore  that  ?'„  is  an  integrating  factor  for  (ii),  and  con- 
sequently that  v^,v^,  .  .  .  ,  v,^  are  integrals  of  (12);  that  is,  the  in- 
tegrals of  (12)  are  the  n  determinants  of 


d^-y^      d"-y^ 


*  '     dx"-^  '     dx''-^ 
d"-y^      d"~y^ 


dx"-^  '        ^^«-3   ' 


'     dx'*-^  *     dx"-^^ 


yn 


J'n- 


y^ 


y. 


[It  is  clear  from  this  that  if  all  of  the  integrals  of  the  given  dif- 
ferential equation  are  regular,  then  all  the  integrals  of  the  adjoint 
equation  are  also  regular.  This  remark  applies  to  all  the  associate 
equations.] 

It  is  known  that  if  (12)  is  the  Lagrangian  associate  of  (11),  then 
reciprocally  (11)  is  the  Lagrangian  associate  of  (12);  and  it  is  evi- 
dent that  if  either  be  in  its  canonical  form,  the  other  will  also  be 
in  its  canonical  form.  Forsyth  shows  now  that  the  dependent 
variable  i>  of  Lagrange's  associate  equation  is  merely  the  last  one  of 
a  set  of  dependent  variables  associated  with  the  dependent  variable 
of  the  given  equation.  These  variables  are  all  transformable  by  a 
substitution  similar  to  that  which  transforms  the  original  dependent 

variable,  viz.,  multiplication  by  some  power  of  -^  ;  and  they  possess 

dx 

the  property  that  all  combinations  of  them,  similar  to  those  by  which 
they  are  constructed,  are  expressible  explicitly  in  terms  of  the  varia- 
bles of  the  set.  The  following  is  Forsyth's  account  of  these  new 
variables: 

Let  jj/, ,  jj/j ,  .  .  .  ,  j«  be  a  set  of  fundamental  integrals  of  the 
given  equation  ;  they  are  of  course  linearly  independent,  and  we 
will  further  assume  concerning  them  (an  assumption  justifiable 
in  the  general  case,  but  not  necessarily  so  in  a  particular  case)  that 
there  exists  no  linear  function  of  them  with  constant  coefficients 
which  is  equal  to  a  polynomial  of  degree  less  than  n  —  i.  Then  of 
course  the  linear  independence  of  the  functions  will  hold  for  their 
derivatives  up  to  the  {n  —  i)"'  inclusive.     Any  other  set  of  funda- 


474  LINEAR  DIFFERENTIAL   EQUATIONS. 

mental  integrals  Fj ,  K, ,  .  .  .  ,  F„  are  linear  functions  with  constant 
coefficients  of  J,,  jj,  .  .  .  ,  j„  .     We  may  write 

(F,,  F,,  .  .  .  ,  F„)  =  S{y,,  J,,  .  .  .  ,  7«), 

where  as  before  .S  denotes  a  substitution  with  non-vanishing  deter- 
minant.    We  have  also 

r/'-F       d'-Y„  d'-Y^ 


dx"-  '      dx'-  '    '  '  '  '      dx' 

for  any  value  of  r.  If  we  retain  this  last  equation  for  values  of  r 
equal  o,  i,  2,  .  .  .  ,  n  —  i ,  we  shall  have  n  sets  of  variables  subject  ta 
the  same  linear  transformation  ;  and  these  variables  are  linearly  in- 
dependent of  one  another,  since  for  the  satisfaction  of  the  differen- 
tial equation  we  need  the  n^^  differential  coefficients  of  the  functions 
^,  but  these  have  been  specially  excluded.  Since  the  n  quantities 
y  are  linearly  independent  they  may  be  looked  upon  as  the  co-ordi- 
nates of  a  point  in  a  manifoldness  of  n  —  i  dimensions;  similarly, 
under  the  hypothesis  made  concerning  the  derivatives  up  to  order 
n  —  I,  each  of  the  n  —  i  sets  of  derivatives,  each  set  being  made  up 
of  derivatives  of  the  same  order,  may  be  looked  upon  as  represent- 
ing the  co-ordinates  of  a  point  in  a  manifoldness  of  n  —  i  dimen- 
sions. And,  since  the  law  of  linear  transformation  is  the  same  for 
all  the  sets,  all  these  points  may  be  taken  as  belonging  to  the  same 
manifoldness.  There  are  thus  11  different  and  independent  sets  of 
cogredient  variables  connected  with  the  single  manifoldness  oi  n  —  i 
dimensions. 

In  the  theory  of  the  concomitants  of  algebraical  quantities  of 
any  order  in  the  variables  of  a  manifoldness  of  ;/  —  i  dimensions,  it 
is  necessary  to  consider  all  the  possible  classes  of  variables  which 
can  enter  into  the  expressions  of  these  concomitants.  Clebsch  * 
has  proved  that  there  are  in  all  n  —  i  different  classes  of  varia- 
bles which  thus  need  to  be  considered,  and  that  li  x^,x^,  .  .  .  ,  x„°, 
/j  >  J2  >  •  •  •  >  J«  ;  -Si ,  -3^5 ,  •  .  .  ,  -«  ;  •  .  .  he  n  sets  of  cogredient  varia- 
bles, the  several  classes  are  constituted  by  minors  of  varying  orders 
of  the  determinant  (itself  an  identical  covariant) 

*  "  Ueber  eine  Fundamentalaufgabe  der  Invariaiitentheorie"  Gottingen,  Abhandlun- 
gen,  vol.  17,  1872. 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 


475 


•^> 

,  x^ ,      .    . 

•       >            -^K 

J. 

A,   .  • 

.       ,            J« 

■^I' 

^,,    .  . 

•     1          '^n 

those  of  one  class  being  minors  of  one  and  the  same  order.  The 
variables  of  any  class  are  linearly,  but  not  algebraically,  independent 
of  one  another,  except  in  the  case  of  the  first  class,  constituted  by 
minors  of  order  unity,  and  the  last  class,  constituted  by  minors  of 
order  n  —  i  (the  complementaries  of  those  of  the  first  class),  in  each 
of  which  classes  the  n  variables  are  quite  independent  of  one  another. 
And  all  similar  combinations  of  variables  are  expressible  in  terms  of 
variables  actually  included  in  the  classes. 

In  connection  with  our  differential  equation  we  have  obtained  n 
different  and  algebraically  independent  sets  of  cogredient  variables ; 
the  functional  derivation  of  the  sets,  one  from  another  in  succession, 
by  the  process  of  differentiation  has  been  excluded  from  any  inter^ 
ference  with  their  algebraical  independence.  We  already  have  one 
class  of  variables,  viz.,  j, ,  J'^,  .  .  .  ,  J„,  analogous  to  the  first  class  of 
algebraical  variables,  and  another  class  of  variables,  viz.,  v^,v^,  .  .  . ,  v^y 
analogous  to  the  {ri  —  i)*^  class  of  algebraical  variables ;  and  the  re- 
lation 

y:^x  +  y-^v^  4-  .  .  .  -^  y^v^  —  o, 

which  is  satisfied,  is  precisely  the  same  as  the  corresponding  relation 
between  the  similar  variables  helping  to  define  the  higher  class 
(Clebsch,  /.  c,  p.  4).  Hence,  from  the  point  of  view  of  purely  alge- 
braical forms,  we  infer  that  the  suitable  algebraical  combinations  of 
the  sets  of  variables,  which  have  arisen  in  connection  with  the  differ- 
ential equation,  are  the  minors  of  varying  orders  of  the  determinant 


7i. 
dx 


dy, 
dx' 


yn 
dyn 

dx 


d"-y,        d^-'y. 
dx"-''      dx"-' 


dx"-' 


4/6 


LINEAR   DIFFERENTIAL   EQUATIONS. 


which,  since  R^  =  o,  is  a  non-evanescent  constant.  These  variables 
may  be  arranged  in  classes  which  may  be  called  linear,  bilinear,  tri- 
linear,  and  so  on.  In  the  case  of  the  algebraical  quantities  it  is  a 
matter  of  indifference  which  set  of  minors  of  a  given  order  be  taken 
to  constitute  the  variables  of  a  class  corresponding  to  that  order. 
Thus  for  the  second  class  the  same  kind  of  variable  is  obtained  by 
taking  the  {x,  y)  minors,  the  {x,  z)  minors,  the  (jj/,  z)  minors,  and  so 
on.  A  difference,  however,  arises  in  the  case  of  the  variables  con- 
nected with  the  differential  equation.  There  are  n  sets  of  linear 
variables  distinct  in  character  from  one  another,  as  the  variables  of 
any  one  set,  say  j/,  j//,  .  .  .  ,  7,/,  though  submitting  to  the  same 
substitution  as  jj,  j.^,  .   .  .  ,  jF„  ,  satisfy  an  entirely  different  differen- 


tial equation.     There  are 
in  character;  thus 


n{)i  —  i) 


sets  of  bilinear  variables  distinct 


yr,  y; 

y:\  y: 


y.,  y. 

v"    V 


are  three  distinct  variables  of  this  class,  subject  to  the  same  law  of 

linear  transformation ;  and  so  on  for  the  higher  classes. 

Most  of  these  must,  however,  be  excluded,  and  of   the  foregoing 

algebraical  combinations  we  must  for  our  purpose  select  only  those 

which  possess,  what  we  may  call,  \.\\&  functional  invariantive  property, 

that  is,  those  which  have  the  invariantive  property  of  reproducing 

dz 
themselves,  save  as  to  a  power  of  -j- ,  after  the  transformation. 

Of  the  n  sets  of  linear  variables  constituted  by  the  several  sets  of 
n  quantities  _)/,  n  quantities  _^',  and  so  on,  only  the  first  set  possesses 
the  functional  invariantive  property,  and  we  already  know  by  (iv),  if 
71  be  the  new  dependent  variable,  that  we  have  the  relation 


(13) 


y  =  la 


-i(«-i) 


n\. 


Of   the   \n{n  —  i)   sets   of   bilinear  variables,  each    set   containing 
^ni^  —  i)  variables,  only  one  possesses  the  functional  invariantive 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 


A7T 


property,  viz.,  the  set  constituted  by  the  \n{ii  —  i)  variables  of  the 
type 

yo.',      fa  I 

y/>        J/3  I 

This  statement  is  readily  verified  by  applying  the  substitution  (13). 
Suppose  /j  denotes  the  original  bilinear  variable,  say  the  one  just 
written,  and  v^  the  transformed  bilinear  variable, 

-7-,         M. 
dz 

dUa 


dz  ' 


u& 


We  have,  since  A  =  ^'-kkn—o^ 

fa      =  ^^a  ^  .        J/3    =   n^  X, 


giving 


Ja    ,       fa 


dUa,  _ 

—r-z'X  —  \{n  —  i)tCaXz'   V,     ti^X 


or 


dz 

ya,       J'a 

j/.   y» 


X  —  ^{n  —  i)ti^Xz'   ^z",     u^X 
dii,^ 


=  r 


dz  ' 
du^ 
dz' 


n^ 


/  =  x-'z'z'  =  vy-^"-'^  =  v„z'-^'^"-'\ 


or,  finally, 

(14) 

It  is  obvious  from  this  that  the  functional  invariantive  property  does. 

not  hold  for  any  bilinear  variable  of  the  form 

ya\     yy 
y?  \     y& 

where  y  and  6  are  respectively  different  from  a  and  yS. 


478 


LINEAR  DIFFERENTIAL   EQUATIONS. 


In  precisely  the  same  way  we  can  show  that  of  the  \n{7i—  \){n—2) 
sets  of  trihnear  variables,  each  set  being  constituted  by  correspond- 
ing minors  of  the  third  order,  there  is  only  one  set  of  which  each 
variable  possesses  the  functional  invariantive  property,  viz.,  the  set 
of  which  the  typical  variable  is 


yy", 


J  y  y 


ya 
yp 
yy 


The  relation  of  transformation  in  this  case  is  easily  seen  to  be 


(15) 


/,  =  vy-'^^"-'\ 


where  z>,  is  the  corresponding  transformed    trilinear  variable.     In 
general,  of  the 

n\ 


P\n-p\ 


sets  of /-linear  variables,  each  set  being  constituted  by  minors  of  the 
J)**^  order,  there  is  only  one  set  which  has  variables  possessed  of  the 
functional  invariantive  property,  viz.,  the  set  of  which 


/.  = 


y-l^ 


y: 

y:^ 


Jl 


/  (/*-I) 


(p-l) 


ypt   yp  1    yp  1 


yp 


p-\) 


is  a  typical  variable.     \(  Vp  denote  the  same  /-linear  variable  asso- 
ciated with  the  transformed  equation,  the  law  of  transformation  is 


<i6) 


/,=  zV^V'+^  + 


=  V.Z 


i/(«-i)4-4/(/-i) 


,/-i/(«-/) 


The  last  set  of  variables  is  that  for  which  p^n  71  —  i  ;  and  the  typical 
variable  of  the  set  is  the  variable  of  the  Lagrangian  adjoint 
equation. 


TRANSFORMATION :    ASSOCIATE   EQUATIONS.  479 

We  see  now  that  there  are  in  all  n  —  i  sets  of  variables  ;  all  the 
variables  in  any  one  set  are  particular  and  linearly  independent 
solutions  of  a  differential  equation  the  dependent  variable  of  which 
is  a  typical  variable  of  the  set.  Hence,  connected  with  the  given 
differential  equation,  there  are  ;/  —  2  other  differential  equations  ; 
these  are  the  associate  eqjiations.  The  n  —  2  new  dependent  varia- 
bles, derived  by  definite  laws  of  formation,  may  be  called  the  asso- 
ciate dependent  variables  ;  and,  calling  them  in  turns  the  associate 
variables  of  the  first,  second,  .  .  .  ,  («  —  2)*''  rank,  the  differential 
equation  of  which  the  dependent  variable  is  the  associate  of   the 

ip  —  i)*''  rank  is  linear  and  of  order  -^ •.     For  the  functional 

transformation  of  the  original  dependent  variable  given  by  (13)  the 
law  of  transformation  of  the  associate  variable  of  the  (/  —  i)'''  rank 
is  given  by  (16) ;  and  if  we  call  two  ranks  complementary  when  the 
sum  of  their  orders  is  11  —  2,  their  associate  variables  of  complemen- 
tary rank  are  transformed  by  the  same  relation,  since  for  such  varia- 
bles the  index:  of  the  factor  power  of  z'  has  the  same  value.  For 
example  :  in  t^  the  power  of  s'  is  —  {n  —  2),  and  in  /„_,  the  power  of 
z'  is  —  {n  —  2),  and  so  t^  and  t„-^,  whose  ranks  (or  orders)  are  re- 
spectively I  and  ?t  —  3,  are  complementary.  The  associate  variables 
may  therefore  be  arranged  in  pairs  of  complementary  rank ;  in  the 
case  of  n  even  there  is  one  dependent  variable  of  self-complementary 
rank.  Each  pair  has  the  index  of  the  factor  power  of  z'  different 
from  that  for  any  other  pair.  The  simplest  case  of  this  arrangement 
is  that  which  combines  in  a  pair  the  original  variable  y  and  the 
variable  /„_i  of  Lagrange's  adjoint  equation  ;  and  the  two  depend- 
ent variables  have  the  same  functional  transformation. 

Leaving  for  the  present  the  general  subject  of  associate  equa- 
tions, we  will  derive  some  properties  of  the  Lagrangian  associate,  or 
adjoint,  equation,  due  in  part  to  Thome  and  in  part  to  Floquet. 
Let  the  given  differential  equation  be 

(17)     P(y)  =  ^^;.  +  P,^^,  +  P,^,+  ---+P.y  =  o. 

This  can  be  placed  in  a  determinate  composite  form  in  the  fol- 
lowing manner :  Writing 


480  LINEAR  DIFFERENTIAL   EQUATIONS. 

construct  the  series  of  linear  differential  equations  of  the  first  order 

^,  =  o,     A^  —  O,     .  .  .  ,     A„  =  o, 
admitting  respectively  as  integrals 

V^,       7\V^,  .     .    .    .    ,        c',7',  .    .    .  T'j,       .    .    .    ,        Z\V^  .    .    .    V„, 

where,  if  j, ,  y^,  .  .  .  ,  j„  are  fundamental  integrals  of  (17),  we  have 

}\  =  ^'.»    J2  =  ^'i/^'.^-^>     •  •  •  ,    J«  =  v.fv^dxfv^dx  .  .  .  fv^dx. 
For  brevity  write 

Vj—  z\i\  .  .  .  t'j,    J  =  I,  2,  ...  ,  n. 
The  value  of  Kj  is  then  given  by 

We  have  identically 

(18)     .  P  =  A,,A,_,.  .  .A,. 

In  fact,  the  expression  on  the  right-hand  side  of  this  equation  is  an- 
nulled by  the  general  integrals  of  the  equations 

^,=  0,  A,=  v^v^,  A^A^—  z\v^v^,  .  .  .  ,  ^„.i^„_2  .  .  .  A^—v,v^  .  .  .v„. 

Now  A^  =  o  IS  satisfied  ior  y  =^  v^,  A^  =  v^v^  is  satisfied  by 
/  =  %\fi\dx,  A^A^  =  vfo^v^  is  satisfied  by  j  =  v^fv^dxfv^dx,  and  so 
on  ;  thus  the  two  equations 

A„A„_,  ...  ^.  =  0     and     P  =  o, 

each  of  order  n,  have  a  system  of  fundamental  integrals  in  common, 

dy  . 
and  consequently,  as  the  coef^cient  of.  -y-^  is  i  in  both  cases,  the  first 

members  of  these  equations  must  be  identical,  for,  if  they  were  not, 
their  difference,  which  is  at  most  of  order  n —  i,  would  be  annulled 
by  n  linearly  independent  functions,  which  is  impossible.     It  follows 


TRANSFORMATION :    ASSOCIATE   EQUATIONS.  48 1 

then  that  (18)  is  an  identity.  In  the  same  way  we  see  that  the 
equation  of  order  n  —  i, 

A{y)  =  A„.,A„_,  .  .  .A,=o, 

admits  the  ;/  —  i  Hnearly  independent  integrals  Jj,  Jj,  .  .  .  ,  J„-i, 
and  consequently  has 

<^iJi  +  Qy.  +  •  •  •  +  C-,J„-i 
as  its  general  integral.     Again,  the  equation 

A{y)  =  Vn 
admits  the  particular  solution  j„,  and  consequently  the  equation 

(19)  A{y)  =  C„V„, 

where  C„  is  an  arbitrary  constant,  admits  the  particular  solution 
€„}'»,  and  so  the  general  integral  of  this  last  equation  of  order  n  —  i  is 

C,}\  +  C  J»  +  •  •  •  +  C„y„ , 

which  is  the  same  as  that  of  the  equation  /*  =  o  of  order  n.  It  fol- 
lows therefore  that  (19)  is  a  first  integral  of  P  =  o.  Writing  (19) 
in  the  form 

V^-^A{y)  =  C„ 

and  differentiating,  we  have 

^[F.-.^W]=o. 

d'y 
Make  the  coefificient  of  -3-^  in  this  unity  by  multiplying  by  V„,  and 

we  have  the  differential  equation 

(20)  Vnf^iV-A{y)]=o, 

whose  first  member  is  identical  with  the  first  member  of  Z'  =  o. 
Writing  F„"'  =  v,  this  identity  is 

(21)  vP{:y)  =  ^^[vA{y)l 


4^2  LINEAR  DIFFERENTIAL   EQUATIONS. 

and  we  see  that 

(22)  V  =   V„-^  = 


v,v„ 


.  ■v„ 


is  an  integrating  factor  oi  P  =  o,  and  consequently  is  the  dependent 
variable  of  the  adjoint  equation 

(23)    P(^)  =^n-  ^—r  {P.V)  +  ^  {P,V)  +...+(_  lyP^V  =  O. 


Consider  in  general  two  differential  quantics 


(24) 


these  will  be  said  to  be  adjoint  differential  quantics,  or  simply  ad- 
joint quantics,  when  the  following  relations  exist,  viz., 

=  /      lY  ^^^  4-  (-  ly-  ^'^"'(^■^^  A-...-  "^^^^-^^^ 

\  /  /V-y<^  I      ^  /  /If<^  —  I  I 


^(7) 


^a^^" 


(25)  i 


SW=(-.)'^ 


+  S.J, 

We  proceed  now  to  establish  an  interesting  relation  existing  be 
tween  a  differential  quantic,  when  in  its  composite  form,  and  its  ad- 
joint quantic.  Suppose  the  quantics  in  (24)  to  be  adjoint,  and  let 
A  denote  a  function  of  x;  we  will  show  in  the  first  place  that  the 
adjoint  quantic  of  S{\y)  is  AS(j).  In  the  first  of  equations  (25)  re- 
place y  by  Xj/,  and  we  have 

the  adjoint  quantic  to  S{Xj/)  is  therefore 


that  is,  \%{y).     Q.  E.  D. 


TRANSFORMATION :    ASSOCIATE   EQUATIONS.  483 

In  connection  with  this  result  we  can  make  the  following  remark: 

We  have  supposed  that  the  coefficient  of  -7-7,  in  Pwas  unity,  and 

so  have  found  that  the  adjoint  quantic  to  P,  which  may  be  written 
in  the  form 

d^y        P^  d"-y       P^  d"-y  P^ 

dx'^~^  P^  dx^—'^  P„  dx'^-^  +  •  •  •  "*"  /^/' 


IS 


d"v        ,        ,         ^'^-^  i  P.   \ 


it  follows  therefore  that  the  adjoint  to 


IS 


„  d"y   ,     „  rtf«-'i/ 

(-  ■)'4?r^  +  (-  ')'-^^  +  •  •  •  +  P-- 

a  result  of  course  immediately  obtained  by  Lagrange's  method. 

Letting  A  still  denote  a  function  of  x,  it  is  easily  shown  that  the 

(  d^y\ 

adjoint  of  the  quantic  5  (A  ~i-:^j  is 

The  differential  quantic 

(d^\  __       d-+^  d-+^-^  d^y 

[dx^j  ~  ^'  dx'^'rk  i-  ^i^_^+^_,  -h  .  .  .  +  ^T-^^ 

has 

or 

'^       '     dx''    ' 


484  LINEAR  DIFFERENTIAL  EQUATIONS. 

as  its  adjoint,  and  consequently,  if  7'(  j)  is  the  quantic  which  has  for 
its  adjoint  \^{y\  (-  ^Y  J^k  PS(  j)]  will  be  the  adjoint  to  T\^j^y 
Now  T{^y)  is  given  by  the  equation 

T{y)  =  S{ly), 

dk  I    d^y\ 

and  therefore  (—  O^"/^  [^S(jF)]  is  adjoint  to  S\\-j^j. 

Suppose,  finally,  that  {S,  S),  {R,  1R)  are  two  pairs  of  adjoint 
differential  quantics,  R  and  IR  being  formed  in  the  same  way  as 
^  and  S;  we  have  now 


RS=  R 


+  R 


-   '  dx"-'-^ 


+  .  .  .  +  R[S^yl 


and  as  it  is  evident  that  the  adjoint  of  a  sum  is  the  sum  of  the 
respective  adjoints,  we  have  at  once  as  the  adjoint  of  RS  the  dif- 
ferential quantic 

that  is,  SIR. 

This  gives  the  following  theorem : 

The  adjoint  quantic  of  RS  is  IRS. 

In  the  same  way  we  can  show  that  the  adjoint  quantic  of  QRS 
is  S1R(Si,  and  so  on  for  any  number  of  differential  quantics.  We 
can  therefore  generalize  the  preceding  theorem  and  say  that — If  a 
differential  quantic  is  composed  of  any  number  of  differential  quantics 
arranged  in  a  given  order,  the  adjoint  quantic  to  the  given  one  is  com- 
posed of  the  corresponding  component  adjoint  quantics  arranged  in  the 
inverse  order. 

The  relation  between  the  integrals  of  the  adjoint  equations 

P{y)  =  o         and         p(j)  =  o 

is  now  easily  found.     Observe  in  the  first  place  that  the  equation  of 
the  first  order 

dy 


TRANSFORMATION :    ASSOCIATE   EQUATIONS. 

has  as  its  adjoint  equation 


485 


dv 
dx 


J^  Jiv  ^=^  O', 


if  the  first  of  these  has  a  solution  y  =  u,  the  second  has  the  solution 
V  =  ti-\     Now  write  /'( j)  in  the  composite  form 

P(j)  =  A„A„_,.  .  .A,, 

where  the  equations 

A^  =  o,      A,  =  o,      .  .  .  ,      A„=  o, 

•of  the  first  order,  have  respectively  the  solutions 


^> .      '^■^', . 


v,v^  .  .  .v„; 


the  equation  P{j)  =  o  then  has,  as  already  shown,  the  following 
system  of  fundamental  integrals,  viz.: 

Ji  =  ^1.      J,  =  "^J'^idx,      .  .  .  ,      y„  =  vjv^dx  .  .  .fvndx. 

From  the  above  theorem  and  the  remark  on  page  483,  the  equa- 
tions 

a«  =  o,      H„-x  =  0,      .  .  .  ,      H,  =  o 

admit  the  respective  solutions 

and  so  |p(w)  =  o,  the  adjoint  equation,  admits  the  fundamental  inte- 
grals 

w,  =  V,r'/v„dx, 

W3   =     V„-'fv,4xfVn-4x, 


(26) 


zv„  =  V„  '/v„dx/v„_,dx  .  .  .  fv^dx, 


which  show  the  relation  between  the  integrals  of  the  two  adjoint 
equations  P{y)  =  o  and  "^{zv)  =  o  when  P^  is  not  equal  zero. 


486 


LINEAR  DIFFERENTIAL   EQUATIONS. 


A  few  Other  properties  of  adjoint  equations,  or  adjoint  quantics,. 
may  be  mentioned.     Writing 


A{y)  =  A„.,A„. 


A., 


d 


we  have  the  identity 

(27)  ^^W=^[^^(j)]; 

if  Ai^y)  be  expanded,  it  is  of  the  form 
d"~^v  d"~'^v 

Equating  the  coefficients  of  the  same  derivatives  in  (27),  we  have  for 
the  determination  of  Z, ,  Z^ ,  .  .  .  ,  Z„_,  the  equations 


dv 


(28) 


+  vL,  =  P^v, 


dx 

d{vL„_^ 
dx 


dx 


+  vL,  =  P,v, 


^- vL,.,=  P„_,v,      "^-^^l^P^v. 

dx 


The  ehmination  of  the  quantities  L  between  these  gives  the  adjoint 
equation 


P(^)  ^  ^^:  _  d-j2{Pf)    ,    d-^{P,v) 


dx.^  dx*' 


dx"-'- 


+  (-    lyPnV   =   O. 


Suppose  the  expression  H(^')  to  be  formed  for  the  adjoint  equation 
in  the  same  way  that  ^(j)  is  formed  for  the  given  equation ;  we 
have  then  the  two  identities 

vP{y)=^-\:c'A{y)l 


givmg 
(29) 


^^  j)  -  yV{^)  =  ^-  \_^>A{y)  -yU{v)] ; 


TRANSFORMATION :    ASSOCIATE  EQUATIONS.  48/ 

that  is,  the  difference 

vP{y)  -  y'^iv) 

is  the  derivative  of  a  differential  quantic  which  is  Hnear  and  homo- 
geneous in  y  and  v  and  whose  coefficients  are  hnear  homogeneous 
functions  oi  P^,  P^,  .  .  .  ,  P„  and  their  derivatives.     We  may  write 

(29)  in  the  form 

(30)  vP{y)  -  y^{2>)  =  —B{y,v), 

The  differential  quantic  B{y,  v)  is  called  by  Frobenius  *  the  beglei- 
tende  bilinear e  Differ entialausdruck  to/-*(jj/);  we  shall  call  it  the 
associate  bilinear  differential  quantic,  or,  when  there  can  be  no 
ambiguity,  simply  the  associate  quantic.  It  is  easy  to  find  now  the 
condition  to  be  satisfied  in  order  that  a  given  differential  quantic 
shall  be  self-adjoint.  Let  F\y)  be  the  given  quantic  and  IP(z')  its 
adjoint;  if  the  coefficient  of  the  highest  derivative  of  y  in  P{y^  is 
P^ ,  then  the  coefficient  of  the  highest  derivative  of  v  in  ^{v)  will  be 
{—lyP^ ;  in  order  then  that  the  quantics  P{y)  and  ]P(z')  shall  be  the 
same  (save  of  course  as  to  the  letter  which  is  used  to  denote  the 
dependent  variable  in  each)  we  must  first  have  that  the  order  n  of 
the  quantic  is  even,  say  ft  =  2v.  This  being  granted,  equation  (30) 
must  have  the  form 

(31)  vP(y)-yP{v)=~B{y,v). 

If  now  we  interchange  y  and  v,  the  left-hand  member  of  this  equa- 
tion changes  sign,  and  so  we  have 

-^B{y,v)  =  -^B{v,y), 
or 

(32)  B{y,  v)  =  -  B{v,  y). 

*  Ueber  adjungirte  lineare  Differentialausdrucke,   G.    Frobenius,    Crelle,   vol.    85, 
p.  185. 


488  LINEAR  DIFFERENTIAL   EQUATIONS. 

Suppose  P^y)  is  the  negative  of  p(^'),  then  we  have 

(33)  'vP{y)  ^yPk-^)  =  i-B{y^  ^) ; 

interchanging  y  and  v  does  not  alter  the  left-hand  member  of  this 
equation,  and  therefore 

B{y,v)  =  B{v,y). 

In  this  case,  of  course,  7i  is  odd.  We  can  state  the  general  result  as 
follows : 

(i)  If  a  differential  quantic  is  self-adjoint  it  must  be  of  even 
order,  and  its  associate  quantic  must  change  sign  when  its  two 
dependent  variables  are  interchanged.  (2)  If  the  adjoint  of  a  given 
differential  quantic  is  the  negative  of  that  quantic,  then  the  common 
order  of  the  two  must  be  odd,  and  the  associate  quantic  must  be 
symmetric  in  its  two  dependent  variables.  It  is  obvious  that  the 
restriction  of  evenness  is  unnecessary  in  the  case  of  a  differential 
equation. 

The  following  remarks  on  adjoint  expressions,  due  to  Halphen,* 
though  reproducing  to  some  extent  what  has  already  been  said,  will 
be  useful  for  future  reference.  Halphen's  notation  is  retained,  at 
a  slight  sacrifice  of  uniformity  of  notation  in  this  chapter,  in  order 
to  facilitate  reference  to  his  important  memoir. 

Denote  by  ^„ ,  ^j ,  .  .  -  ,  y^,  y^,  -  -  •  given  functions  of  the  inde- 
pendent variable  x ;  by  y  and  r/  indeterminate  functions  of  x ;  con- 
sider the  differential  quantics 

(^{y)  =  ^o/"'  +  ^/^,y"-'  +  ''^'\~'\./"-''  +  .  .  •  +  n^n-^y'  +gny, 

r{rf)  =  y,rf^^  +  ;^^,^(«-)  +  "^'1^}  y^rf^-^)  +  •   •  .  +  ny„.,r^'  +  r«V- 

These  two  linear  quantics  are  said  to  be  adjoint  to  one  another  if 
there  exists  a  third  bilinear  quantic  B{y,  ij)  which  is  linear  and 
homogeneous  both  with  respect  to  y  and  its  derivatives  up  to  the 

*  Sur  un  problhne  concernant  les  equations  diff/rentielles  lin^aires.      Par  M.   G.-H. 
Halphen.     Journal  de  Math6matiques  pures  et  appliquees,  4'"«  Serie,  t.  i.  p.  ii. 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 


489 


order  n  —  \,  and  with  respect  to  /^  and  its  derivatives  up  to  the  same 
order,  and  which  is  such  that  we  have  identically,  that  is,  whatever 
be  the  functions  7  and  /;, 


(34) 


vG{y)  +  (-  iY-yi\v)  =  B\y,  /;), 


accents  as  usual  denoting  differentiation  with  respect  to  x.  This 
relation  completely  determines  one  of  the  quantics  G{^y'),  -^(v)  when 
the  other  is  arbitrarily  given.  The  relation  between  these  quantics 
is  expressed  by  either  of  the  following  systems  of  equations  : 


(35)     \ 


Y-i      gi       ~i>\     I  <so 

Y^  <53  I  3^>3       lyg\      \gii 


g,  =  Yo 

<^i  =  —  r.  +  Ko' 
g.  =  Y^-  2r/  +  Yo" 
g^  =  — r3+3r/-3r/'+ro 


The  law  of  these  equations  is  obvious,  and  the  two  adjoint  quantics 
have  the  forms 


(36)     \ 


G{y)  =  (ko j)""  -  nir^yY"-^^  +  '-^j^Cn^^y^-^)  +  .  .  . 

+(-i)"+y«j, 


The  two  quantics  G{y)  and  r{ij)  being  formed  in  this  manner,  (34) 
is  satisfied  when  the  associate  quantic  B{y,  ij)  has  either  of  the  forms 


(37)     \ 


+  (-i)«A.-./  +  (-i)"+'/?«-xj, 


490 
where 


(38) 


(39) 


LINEAR  DIFFERENTIAL  EQUATIONS. 

A  =  {g.vi  -  ^g^V, 

N,/   ,    K'^— 0/       N/       n(n—i)(n—2) 
A  =  (.^o7)'"  -  «(.^.V)  '  +  ^YTy-^  (.^,v)'  -        ^.,.3      V37-. 

•••••••••  •••>! 

-^0   =    Ko  J» 

.,,  ,   ^^0^— 0,       X,      n(n—\)(n  —  2) 


1-2 


I  -2 


In  order  that  two  quantics,  say  P{j/)  and  p(t'),  may  be  self- 
adjoint,  we  have  seen  that  they  must  be  of  even  order,  and  that  on 
interchanging  y  and  v,  B{y,  v)  must  change  sign.  As  illustrations 
take  the  cases  of  n  —  ^  and  n  —  6,  and  use  (37),  For  «  =  4  the 
quantic  P  is 


and  its  adjoint  expanded  is 


^V        ^d\>   ,   (^         d'P\  d\>      I  dP,  .      d'P.\dv 


^'       ^  dx  +  ^  dx""  I  dx 

dP,  .  d'P„    d'P: 


+  ^^'        dx  +  dx'        dx' 


IK 


The  conditions  that  («)  and  (o')'  shall  be  the  same  are  at  once  found, 
by  comparing  coefficients,  to  be 


P  =0     P  —  P       P 


d^, 
dx 


P  —  P  . 


TRANSFORMATION :    ASSOCIATE  EQUATIONS.  49 1 

The  associate  quantic  is 

(/i)     B^y,  V)  =  vf'  -  [z/  -  P.i^y"  +  \_v"  -  P,v'  +  (/>,  -  P:)v']/ 
-  [v'"  -  Py  +  {P.  -  2P:)v'  -  {P,  -  P:  +  P/>]  J  ; 

arranged  according  to  v  this  is 

(^y     -  v"'y  +  v'\P,y  +  j']  -  z.'[(/',  -  2P/)j  +  P.  j'  +/'] 

+  v\{P,  -  p;  +  p/0/  +  (^.  -  AOj'  +  PJ'-\-y"'\ 

In  (^)  interchange  j  and  v  and  we  have 

{§)"     'v"'y  -  v"\y'  -  P^y-]  +  v'[y"  -  PJ  +  (P,  -  /^/)j/] 

-  v\y"'  -  P^y"  +  (P,  -  2P:)y'  -  {P,  -  P/  +  P/>]. 

That  (/5)'  and  {ft)"  may  have  opposite  signs  we  must  obviously  have 

y-P,y  =  P,y+y\     .:P,  =  o; 

using  this,  we  have  next 

y"^P.y  =  P.y+y'\    .:P,  =  P,; 
again, 

-  /"  -  PJ  +  (^s  -  P:)y  =  -  y'"  -  p.y  -  {P.  -  P:)y, 

giving 

and  finally  P^  =  P^.     These  are  the  same  results  as  found  by  direct 
comparison  of  {a)  and  (or)'. 

Take  the  case  now  of  n  =  6.  From  the  general  form  of  the  ad- 
joint equation  it  is  obvious  a  priori  that  the  coefificient  of  the  second 
highest  derivative  must  be  zero,  i.e.,  Pj  =  o ;  the  sextic  can  then  be 
written 

iv\  ^-^.p^^-p'^-^-^.p^'-A-p'^lA-Pv 


492 


LINEAR  DIFFERENTIAL   EQUATIONS. 


the  adjoint  sextic  is 


d'v 


d'v 


^y^      Tx'^^^lx^- 


+ 


dP. 


d'P. 


d'p: 


dv 


"  dx  '^  ^  dx""       '^  dx^  J  dx 
dP,  ,  d'P,      d'P,  .  d*Pfi 


dx        dx' 


dx^        dx" 


In  order  that  {y)  and  {y)'  shall  be  the  same  we  must  have,  as  is 
■easily  seen  by  comparing  coefficients, 


^'  dx'         ' 


dP,  _  d_^ 
dx         dx'  ' 


/*, ,  /*4,andPg  remaining  arbitrary.  The  same  results  of  course  must 
be  obtained  by  considering  the  associate  quantic  B{^y,  z') ;  this,  which 
is  easily  found,  is 

(6)  vf  -  v'y-  +  \v"  +  p^vy  -  [./-  +  p^o'  -  {p,  -  p;y^" 

+  \ir  +  p^v"  -  (^3  -  2P:y  +  (p,  -  p:  +  p:y^y 

-  \:o^  +  p^"  -  {p,  -  zp.:)v"  +  (p,  -  2p:  +  zp:')v' 

-{p.-p:  +  p:'-p:">']y; 

arranged  according  to  v  this  is 

^Sy     -  vy  +  v-y'  -  v"'\P^y-\-y"^  +  v"\{P,  -  iP:)y  +  P^y'  +  j'"] 

-  v'\{p,  -  2p:  +  2,p:')y + {P.  -  2P:)y'  +  p.y  +yT 
+ z{(p,  -  p:  +  7^3''  -  /'/'Oj + {P.  -  p:  +  p:')y' 

^{P.-P:)y"-\:P.y"'-\-f\ 

In  ((J)  interchange  y  and  ?;  and  we  have 

{8)"     v^y  -  v-y'  +  v"\y"  +  P,j/]  -  v"\y"  +  /'.j.'  -  (P,  -  P:)A 

+  t''[y^  +  p,/'  -  (P3  -  2P;)/  +  (P.  -  p/  +  p;>] 

-  ^'b^  +  ^./"  -  (^3  -  i>p:)y"  +  {P.  -  2p:  +  3/'.")/ 

-{p.-p:  +  p:'-p:")yy 


TRANSFORMATION:    ASSOCIATE  EQUATIONS. 


493 


In  order  that  {S)'  and  ip)"  shall  have  opposite  signs  we  must 
have,  as  is  readily  seen, 

^=  ~      dx  '        '  ~  dx        dx'  ' 

In  the  case  of  the  quartic  (assuming  always  P^  =  o)  replace 
P^ ,  /*3  by  6P^ ,  4/^3 ,  and  in  the  case  of  the  sextic  replace  P^,  P^,P^,P 
by  1 5/^2 ,  20P3 ,  1 5/^4 ,  6P^ ;  that  is,  multiply  the  coefficients  of  the  two 
quantics  by  the  corresponding  binomial  coefficients.  The  conditions 
that  these  quantics  shall  each  be  self-adjoint  are,  for  the  quartic, 


(40) 

for  the  sextic, 

(41) 


'        2  dx        "' 


p. 


idP^ 
2  dx 


=  0, 


"       2  dx        3   dx"^ 


Forsyth  in  his  memoir  already  referred  to  gives  the  forms  of 
certain  invariants  connected  with  the  general  linear  differential  equa- 
tion ;  among  these  are  two,  denoted  by  ©3  and  ©^respectively,  which 
have  the  forms 


(42)    \ 


^'  '       2  dx' 

@    —  /»    ± 1  J ^  ?  —  2.  ? L \ ^  p  Q 

"         "      2  dx       y    dx''       7  dx^        y    ?i -\-  i      '    " 


Comparing  these  with  (40)  and  (41),  it  is  seen  at  once  that  the  con- 
ditions to  be  satisfied  in  order  that  a  quartic  or  sextic  linear  differ- 
ential quantic  shall  be  self-adjoint  are,  for  the  quartic, 


for  the  sextic, 


©3  =  0; 
©3  =  0,     0,  =  o. 


494  LINEAR  DIFFERENTIAL  EQUATIONS. 

In  the  case  of  the  octic 

(■,o,/'„/'..../'.|^,i)^ 
the  conditions  for  self-adjointness  are 


'  2    dx  "' 


or 


and 


^^  2    dx^  I    dx""    ~  "' 


"■  2    dx^  2    dx'  "' 


''       2  dx        4    dx""         2    dx" 
In  terms  of  Forsyth's  invariants  these  are 

©3  =  o,     ©5  =  o,     ©,  =  o. 
The  general  form  of  ©,  is 

'  '        2  dx'^  22    dx"^         II   ^^'         33   ^;i;^         44  ^;i;' 

7     P. 


II  ;z-f- 


-jf(ii.  +  3i)(2P.-5f)  +  5(i5^^  +  4i) 


^Z*. 


II   ;^  +  I    /  '^  dx'  \   '   ^    dx  I       ^  dx  dx 


7  3«  +  4L^. 
I-  I   P  ^^-'^ 

1 1 5  5«'  +  6048 w  +  6909 


^'/l  ^P, 


+  p:  & 


22{n  +  i)* 


I 


TRANSFORMATION :    ASSOCIATE  EQUATIONS. 


495 


Mr.  G.  F.  Metzler  has  put  this  invariant  in  the  following  form, 
■^vhich,  for  the  present  purpose  at  least,  is  rather  more  desirable  than 
the  above  : 


^  ~       '  2   dx  "^    22     dx'  1 1    dx' 


35  d'P. 
33  dx' 


7_d^ 
44  dx" 


7     P. 


\\{n  +  I) 


(33^^  +  93)0. 


-?_p.0  38_5«'+  I728;/+I9I9 


10  P. 


^3 


22 


35 -y— -7- +  21 


(;.+  ir 


^'P„ 


0, 


^/^-   ^J 


The  subject  of  adjoint  quantics  will  be  taken  up  again  after  an  ac- 
count of  the  invariantive  theory  has  been  given  ;  enough  has  already 
been  said,  however,  to  show  that  a  differential  quantic  is  self-adjoint 
when  certain  of  its  invariants  vanish,  and  it  is  easy  to  see  that  in  any 
case  three  of  these  invariants  must  be  ©3,  ©j,  and  ©,.* 

An  interesting  theorem  due  to  Appell,  and  closely  related  to  the 
invariantive  theory,  may  be  merely  stated  here  without  proof ;  the 
reader  is  referred  to  Appell's  paper  in  vol.  90  of  the  Comptes  Rendus, 
p.  1477.     Given  the  equation 


dx^^     ' 


y 


ix""- 


P. 


d^' 


y 


dx''' 


+ 


+  Pny  =  o', 


let  jj ,  y^,  .  .  .  ,  y„  be  a  set  of  fundamental   integrals;    Appell's 
theorem  is : 

Every  integral  algebraic  function,  F,  oi  y^,  y^,  .  .  .  ,  y„  and  the 
derivatives  of  these  functions,  wJiich  reproduces  itself  multiplied  by  a 
constant  factor  other  tJian  zero  when  we  replace  y^^  y^,  .  .  •  ,  yn  ^y  t^i^ 
elements  of  anotJier  fundamental  system,  is  equal  to  an  integral  alge- 
braic function  of  the  coefficients  of  the  differential  equation  and  their 
derivatives  multiplied  by  a  power  of  e-f^^'^^. 

*  Mr.  Metzler  has  proved  that  the  condition  for  self-adjointness  in  general  is  that 
the  invariants  0  with  odd  suffixes  must  all  vanish.  The  proof  is  rather  too  long  and 
complicated  to  give  here. 


CHAPTER   XIV. 

LINEAR   DIFFERENTIAL   EQUATIONS   WITH   UNIFORM   DOUBLY- 
PERIODIC   COEFFICIENTS. 

Before  taking  up  the  subject  of  the  integrals  of  these  equations 
it  will  be  convenient  to  recall  a  few  points  in  the  theory  of  doubly- 
periodic  functions.  These  functions  are  all  constructed  by  aid  of 
the  element  function  ^X-^')  (o^»  i^  a  he  a.  constant,  8^{x  —  a)).  If  go 
and  oj'  denote  the  periods  of  the  doubly-periodic  function,  then  we 
know  that 


27r;a-      ir;co 


e,{x  +  Go)  =  -  e^x),    e,{x ^oo')  =  -e    <-  -  „  e^x). 

Another  element  function  which  plays  an  important  part  in  the 
theory  is  the  logarithmic  derivative  of  6^ ,  viz., 

For  X  =  0-d^{x)  has  a  simple  zero,  and  so  Z{x)  has  a  simple  pole. 
Inside  each  parallelogram  of  periods  the  doubly-periodic  function 
will  have  a  certain  number  of  zeros  and  a  certain  number  of  poles 
(account  being  taken  of  the  orders  of  multiplicity).  Now  it  is  a 
known  theorem  that  the  number  of  zeros  inside  a  parallelogram  of 
periods  is  equal  to  the  number  of  poles.  Let  this  number  be  ;//,  and 
let  «j ,  ,  .  .  ,  rt,„  denote  the  zeros  and  a-, ,  .  .  .  ,  nr,„  the  poles  inside 
a  parallelogram  of  periods ;  then  by  another  theorem  we  have 

m  nt 

(1)  ^ (li  —  ^  ai  ■=^  }xoo -\- p^ 00  , 

I  I 

where  yu  and  /^'  are  integers.  Now  a  doubly-periodic  function  hav- 
ing 00  and  00'  as  periods  is  given  by 

,  V  - ^'^'^-"'  Bix  —  a\  .  .   .  6 Ax  —  a,,') 

(2)  e       " -^  . 

^  ^  Oix  -  a,)  .   .  .   Six  -  a,„) 

496 


DOUBLY-PERIODIC    COEFFICIENTS.  497 

This  obviously  has  «, ,  c  .  .  ,  «,„  as  zeros  and  a^,  .  .  .  ,  a,„  as  poles. 
Further,  cj  is  a  period  ;  in  fact  the  exponential  is  unaltered  by  the 
change  of  x  into  x  -\-  go,  and  each  function  6^  merely  changes  its  sign 
to  minus — there  are,  however,  an  even  number  of  these  functions, 
and  so  the  sign  remains  unaltered.  Again,  go'  is  a  period  :  changing 
X  into  X  -\-  go'  reproduces  the  exponential  multiplied  by  the  factor 

2/u.'7r/<o' 

e       ^"; 

6,{x  —  a^)  is  changed  into  —  ^"V  <-'''''  6^{x  —  rt-,),  and  so  for  all 
of  the  other  functions  6^ .     The  whole  function  is  then  multiplied  by 

^^' /     /III  \ 

(-  H'w   +  «i  +  .   .   .  +  "m  -  a,  -  ...  —  am)  ■ 

Another  representation  of  the  doubly-periodic  function  is  due  to 
M.  Hermite.  Suppose  a,  b,  .  .  .to  be  the  poles  of  the  function  in- 
side a  given  parallelogram  of  periods,  and  o',  /?,...  to  be  their 
respective  orders  of  multiplicity;  the  doubly-periodic  function /(.«•) 
is  then  given  by  the  formula 

(3)  f{x)  =  A,Z{x  -a)-  A,Z'{x  -«)+... 

'     I  .  2   .   .   .   (or  —  l)  ^  ^ 

+  B,Z{X  -b)-  B,Z'{x  -/;)  +  ... 

'      I  .  2    ...(/?—  l)                 ^  ' 

+ 

+  c, 

where  C  is  an  arbitrary  constant,  and  where  the  sum  of  the  residues 
A^,  B^  ,  .  .  .is  zero,  viz., 

(4)  yi,  +  ^,+  .  .  .=o. 

This  need  not  be  verified  here.  Another  and  very  important  class 
of  doubly-periodic  functions  are  the  doubly-periodic  functions  of  the 
second  kind,  as  they  are  called  by  Hermite.  Denote  by  0  such  a 
function  with  periods  go  and  go'  as  before.  Then  0  satisfies  the  rela- 
tions 

e{x  +  cy)  =  s^e{x),     @{x  +  go')  =  s/9{x), 


49^  LINEAR   DIFFERENTIAL   EQUATIONS. 

Avhere  jt,  and  j/  are  constants.  It  is  easy  to  form  such  a  function  as 
this  by  aid  of  the  function  ^^x),  and  at  the  same  time  to  let  the  con- 
stants s^  and  s^  take  any  value  we  please.     Write,  viz., 

where  /  and  q  are  arbitrary  constants.     We  have  now 

(6)  @{x  -\-Go)  =  e^'^Q{x\     Q{x  +  a?')  =  ^         '-'''  ©W- 
Now  since/  and  q  are  arbitrary  constants,  we  may  write 

(7)  ^'-  =  s.,     e'-'^^'  =  s:, 

where  5,  and  s^  are  equally  arbitrary  constants.  Suppose  now  that 
among  the  infinite  number  of  values  of  the  logarithms  of  s^  and  s^' 
we  choose  arbitrarily  two  values  which  we  will  denote  by  Lg^j  and 
Lg5/  ;  denoting  now  by  m  and  ;//  two  arbitrary  integers,  we  have, 
from  (7), 

[  poo  =  Lg5j  +  2/;/7r?, 

(R\  -{  27ti 

^  ^  \  pGo'  -\ q  =  Lg^/  -|-  2m7TZ, 

\  GO 

from  which  follow 

Lg^j       2m'7ti 


(9) 


/ 


q  =  — .  [cipLg^/  —  oo'l^gs^  -{-  moo  —  in' go' 


Among  the  systems  of  values  of  /  and  q  given  by  these  equa- 
tions there  is  one  only  for  which  the  point  q  lies  inside  the  parallel- 
ogram of  periods  having  the  origin  as  one  vertex  and  otherwise 
determined  by  the  periods  oo  and  oo' ;  this  system  of  values  of/  and 
q  will  be  the  one  employed  in  all  that  follows. 

The  function  ©{x)  so  formed  admits  the  multipliers  s^  and  s^' 
with  respect  to  the  periods  go  and  oo',  and  inside  the  parallelogram 
of  periods  considered  has  the  simple  zero  x  ^=  q  and  the  simple  pole 
^  =  o.     The  case  of  ^  =  o  must  of  course  be  excepted,  as  in  this 


DOUBLY-PERIODIC  COEFFICIENTS.  499 

■case  ©{x)  reduces  to  the  exponential  c*""  and  has  no  zero  or  pole  in 
the  parallelogram.  A  doubly-periodic  function,  which  is  to  be  con- 
sidered presently,  has  periods  co  and  od'  and  poles  a  -\-  q,  a,  b,  .  .  .  o{ 
orders  of  multiplicity  respectively  equal  to  i,  a  -\-  i  —  2,  §,  .  .  . 
Denoting  such  a  function  by  ^,4,  we  have,  from  (3), 

(10)  ^a{^)  =A,Z{x-a)+A,Z'{x-a)-^  .  .  .  -{-A,+  ,-_,Z^<^+^--^Hx-a) 

J^B,Z{x-b)^B,Z\x-b)^  .  .  .  +B^Z^-^{x-b) 
-}-...  + 3fZ{x-a-q)-\-C, 

where  A,  B,  .  .  .  C  are  constants  and 

(11)  A,  +  B,  +  .  .  .-\-M=o. 

This  function  can  be  given  in  terms  of  elliptic  functions  by  aid  of 
the  relation 

(12)  Z{x-a)  -  Z{x)  =  '"^ ^z(^-a)-Z  (^\ 

The  truth  of  (12)  is  easily  seen:  in  fact  both  members  of  the 
equation  have  the  same  periods  00  and  00' ,  the  same  simple  poles 
O    and   a,    and    the    same   residues ;    finally,  they   are    identical  for 

2 
tion 


X  =  — .     We  can  obtain  from  (12)  (or  can  verify  directly)  the  rela- 


(.3)  z'(^-«)=--_!_  +  z'f^Y 

sn{x—  a)  \  2  ) 

From  this  last  equation  by  differentiation  we  get  the  values  of 

Z"{x-a\      Z"\x-a\      .  .  . 

Form  in  the  same  way  the  values  of 

Z{x-b),      Z\x-b),      ... 


500 


LINEAR  DIFFERENTIAL   EQUATIONS. 


Eliminating  now  all  of  these  quantities  from  (lo),  and  denoting  by- 
C  the  new  additive  constant,  we  have 


(14)     ^ik{x)  =  A, 


A. 


sn  X  sxiix  —  a)        sv^{x  —  ci) 
<^"+'-4  I 


sn  d 


sn  X  sn(;r  —  d) 


.-{-M 


sn(«  +  ^) 


sn  X  sn(;ir -—a  —  g) 


c: 


For  the  complete  determination  of  the  function  ^ik{x)  in  any- 
particular  case  it  is  only  necessary  to  determine  the  constants 
A,B,...,  M,  C. 

One  more  preliminary  remark  may  be  made  before  proceeding 
directly  to  the  investigation  of  the  integrals  of  the  given  dif^erentiaL 
equation.     The  function  Z{x)  satisfies  the  relations 

(15)  Z[x  +  a?)  =  Zix),       Z{x-\-oo')  = f-  Z{x). 

GO 

Form  now  the  function 

(16)  mx  -\-  m'Z{x  —  a),  =  (p{x), 

where  m,  m' ,  and  a  are  constants.  If  we  change  x  into  x  -\-  00  and 
X  -\-  go'  successively,  this  function  (p{x)  will  receive  increments  A(p{x)- 
and  A'(j){x)  defined  by  the  equations 


(17) 


'  A(p[x)    =  moo, 

A  chlx)  =  moo m  . 

00 


We  can  clearly  choose  the  constants  ;;z  and  ;;/  so  that  these  incre- 
ments shall  take  any  values  we  please.  Suppose  then  we  construct 
a  function  /i  of  the  form  (16)  such  that 

(18)  A  1.1=  I,       A').i  =  o; 
also  a  function  fx'  of  the  same  form  such  that 

(19)  ^/i'  =  o,      A'/a'  =  r. 


DOUBLY-PERIODIC  COEFFICIENTS. 


501 


These  functions  /<  and  /  will  have  inside  the  parallelogram  of  peri- 
ods the  same  simple  pole  x  —  a  2.s  that  possessed  by  the  function 
Zi^x  —  a)  from  which  they  are  derived. 
If  now  we  write 


(20) 


f^o 


I,      /X,  =  My      Mi  = 


>         *     • 

I   .  2 


M»  = 

Am'-  0 


{m  -  n-\-  i) 


I  .  2 


I  .  2 


(;/'  -  ;/  +  I) 


I  .  2 


we  have  obviously 

(^  +  i)//(/^  -  I) 


<2I) 


^/^«  = 


(a/  —  ?z  +  2) 


I  .  2   .   .  .  « 


(^  _  ;^  +  l) 


I   .  2 


M»-i 


A'fA^  =  o, 


a  series  of  relations  similar  to  those  obtained  in  Chapter  III  (equa- 
tions 84-86)  for  the  functions  d^  ,  0^ ,  .  .  .  ,  6^,  .  .  .  .  Every  poly- 
nomial, integral  in  both  pi  and  ju',  can,  considered  as  a  function  of  yu, 
be  written  in  one  way  only  in  the  form 

A,^,  +  A,f4,  4-  .  .  .  , 

where  the  coefificients  A  are  polynomials  in  pi'  having  the  form 

A,  =  Bi,^:  +  ^,,///  +  .  .  .  , 

Bi^,  Bi^,  .  .  .  being  constants.  Any  such  polynomial  then,  say  77, 
can  in  one  way  only  be  placed  in  the  form 


(22) 


n  =  :sB^^>pio.M'a 


502  LINEAR  DIFFERENTIAL  EQUATIONS. 

By  aid  of  (21)  we  derive  at  once 

o,  a' 


(23) 


Consider  for  a  moment  the  polynomials 
(24) 


^     t-i  Im    ^  I 


^  a.  a' 


a  -{-  a'  =  \  —  I  —  f, 


which  are  of  the  same  form  as  77.     We  have  at  once 

a,  a' 


(25) 


The  expressions  for  ^HaI  and  A' ETk  are  not  needed;  their  forms 
are,  however,  obvious.  The  expressions  in  (25)  are  again  polynomi- 
als in  /x  and  jx'.     Suppose  the  condition 

(.26)  A  ^^ji  =  AH \k 

is  to  be  satisfied ;  from  (25)  we  must  then  have 

\.2/J  A'a— I,  a'    ^^=    /If  o,  a'-i  • 

We  can  now  determine  a  polynomial  of  order  A.  —  /  in  //  and  fx\ 


say 
(28) 


n^"l    =    :2  Baa'MaM'o^')  «  H"  «'  <  ^, 


o,  a' 


such  that  its  variations 


(29) 


An''"         =^^„a'i"a-./a', 


A'n  ^l,    =   2  Baa'  Ma  M^a'-i  > 


DOUBLY-PERIODIC  COEFFICIENTS.  50$ 

shall  be  respectively  equal  to  Hl'l,  'E''"'k\  for,  in  order  that  these  re- 
lations may  hold  we  must  have 

(^3*-*/  Liaa.'    —    /Ifa— 1,  o' >  J^aa'    —    /If  a,  a'  — i  • 

Now  if  the  functions  H,  E'  have  been  determined  in  such  a  way 
that  equation  (26)  is  satisfied,  it  follows  immediately  from  equation 
(27)  that  equations  (30)  can  also  be  satisfied. 

We  will  take  up  now  the  investigation  of  the  integrals  of  the 
linear  differential  equation  with  uniform  doubly-periodic  coefificients. 
We  have  seen  in  the  case  of  the  linear  differential  equations  already 
studied  that  there  always  exists  at  least  one  integral  of  the  equa- 
tion which,  when  the  variable  travels  round  a  critical  point,  is  changed 
into  itself  multiplied  by  a  constant  factor,  the  factor  being  a  root  of 
the  characteristic  equation  corresponding  to  the  particular  critical 
point ;  or,  in  other  words,  the  effect  of  imposing  upon  this  integral 
the  substitution  corresponding  to  the  critical  point  considered  is  to 
multiply  the  integral  by  a  constant  which  is  a  root  of  the  character- 
istic equation  of  the  substitution.  The  uniform  coefificients  of  the 
equation  are  unaltered  when  the  variable  turns  round  the  critical 
point.  In  the  equation  with  uniform  doubly-periodic  coefificients 
these  coefficients  are  equally  unaltered  when  the  variable  x  is  changed 
into  X  -\-  ot)  ox  into  x  -(-  00' .  The  question  now  naturally  arises  in 
this  case  as  to  whether  or  not  the  equation  possesses  an  integral 
which  is  multiplied  by  a  certain  constant,  say  s,  when  x  changes 
into  X  -\-  00,  and  by  a  certain  constant,  say  s',  when  x  changes  into 
X  -\-  go'.  Corresponding  to  the  changes  of  x  into  x  -\-  co  and  x  -\-  go^ 
respectively,  we  have  the  substitutions  5  and  S' ;  and  so,  if  such  an 
integral  as  we  have  described  exists,  we  might  expect  by  analogy  to 
find  the  multipliers  s  and  s'  as  roots  of  the  characteristic  equations 
corresponding  to  the  substitutions  5  and  S'.  Picard  has  shown  that 
every  linear  differential  equation  with  uniform  doubly-periodic  co- 
efficients and  possessing  only  uniform  integrals  has  always  at  least 
one  integral  which  is  a  doubly-periodic  function  of  the  second  kind 
whose  multipliers  s  and  s'  are  roots  of  the  characteristic  equations 
corresponding  to  the  substitutions  5  and  S'.  That  is,  the  equation 
always  has  at  least  one  integral  such  that  when  x  is  changed  into 
X  -{-  GO  the  integral  is  multiplied  by  s,  and  when  x  is  changed  into 


504 


LINEAR   DIFFERENTIAL   EQUATIONS. 


X  -\-  00  the  integral  is  multiplied  by  5'.*  It  has  been  supposed  that 
the  integrals  of  the  differential  equation  are  all  uniform  ;  whether 
or  not  the  integrals  possess  this  property  can  always  be  ascertained 
by  developing  them  in  the  form  of  series.  In  what  follows  we  will 
assume  that  the  integrals  always  satisfy  the  condition  of  being  uni- 
form. We  have  seen  in  Chapter  III  that  it  is  possible  to  determine 
a  system  of  fundamental  integrals 


fjii,   •  •  •  ,  yu,,    ^21,  •  •  .  ,  y^i^, 

\jy]'3^ii>       •     •     •     >     '^nn-^1      -^21  >       •     •    •     »      ^2/«2  ) 

L 


7Ai 


such  that  the  substitutions  5  and  S'  shall  take  the  canonical  forms 


y.k,  •  . 

•  ,  yik,  '  • 

■  ;  -Jiji*,  .  • 

•,     S,{^,,+  Va),    .    . 

^ikj  •  • 

•     >     ^ikt      •   • 

•  )  s^^ik,   •  - 

.  ,      ^.(^/*  +  ^ik),    '   ' 

y.k,  .  . 

•  ,  yik,  . . 

'  ;  ^,>ii.  •  • 

.  ,  s/iya  +  r,,), .  . 

^i*,  •  . 

.    ,    2i/i,     .  . 

•  ',  ^i  ^ik ,  •  • 

•  ,    ^ii^ik  +  ^'ik),  •  • 

(32)  5 


(33)  5' 


where  s^,  s„,  .  .  .  are  distinct  roots  of  the  characteristic  equation  cor- 
responding to  the  substitution  S,  and  s/,  s^\  .  .  .  are  distinct  roots  of 
the  characteristic  equation  corresponding  to  S',  and  where  Vn, ,  Y'n, 
are  linear  functions  of  the  integrals  y  whose  first  sufifix  is  less  than 
i,  etc.  The  integrals  now  being  supposed  to  be  chosen  so  that  (32) 
and  (33)  are  satisfied,  let  us  consider  the  class  jj,,  .  .  .  ,  yik,  ...  If 
now  we  change  x  into  x  -\-  oo  and  x  -\-  go',  we  have  the  partial  sub- 
stitutions 


(34)    o-  =  |/,4,     .  .  .  ,    J/,-,, 

(35)  <^'  =  \y:k,     " ' ,  yik, 


;  -j/ji* ,  . . . .  s/{j'ii  +  V'ik), . . .  I  , 


*  Picard:  Sur  les  Equations  diffc'rentielles  Im^aires  h  coefficients  doiibkment  piriodiqucs, 
Crelle,  vol.  go,  p.  281.  See  also  Floquet:  Sur  les  Equations  diffdrentielks  lindaires  a 
coefficients  doublement  p^riodiques.  Annales  de  rEcole  Normale  Sup6rieure,  1884.  The 
student  should  also  consult  Hermite:  Stir  quelques  applications  des  Fonctions  FJlip- 
tiqites.     Gauthier-Villars,  Paris,  1885. 


DOUBLY-PERIODIC  COEFFICIENTS.  505 

where,  by  hypothesis, 

(36)  Y,,  =  2  dl  j,„, ,     F',,  =  2  b'l  J,™ , 

I,  m  I,  m 

where  the  summations  extend  to  all  values  of  the  first  index  /  which 
are  less  than  t,  and  to  the  corresponding  values  of  the  second  index 
m.  Suppose,  for  example,  /  =  r,  where  v  is  less  than  t;  then  the 
corresponding  values  of  m  will  be,  from  (31),  m  =  i,  2,  .  .  .  /^. 

Since  we  have  the  relation  55'  =  S'S,  we  must  also  have,  from 
(34)  and  (35), 

(37)  aa'  =  a-'a-. 

This  relation  can  be  expressed  in  another  form  by  aid  of  equa- 
tions (36).  Consider  the  function  y,/, ',  the  substitution  a  changes  it 
into 

■(38^  cry,,  =  siy,,  +  F,,)  =  s,\_y,,  +  2^2  J/.J, 

/,  m 

and  the  substitution  cr'  gives 

(39)  ^'yik  =  s/{y,,  +  F;,)  =  ./[  J.,  +  2  b%y,^. 

I,  m 

If  now  we  first  make  the  substitution  (t  and  then  follow  it  by  (t', 
i.e.,  if  we  make  the  substitution  ccr',  we  find  at  once 

(40)  cTCT'y,,  =  s,s/ly,,  +  Y,,  +  F',,  +  2\  a^l  2b'-' y,„,  \\ 

i,  m  I',  m' 

where  the  summation  with  respect  to  /'  and  in'  extends  over  all 
values  of  I'-  which  are  less  than  /,  and  over  the  corresponding  values 
of  m' .  The  substitution  crV  applied  to  the  same  function  yi^  will 
clearly  only  have  the  effect  of  interchanging  the  letters  a  and  b  in 
(40).     Now  since 

0-0-%  =  cr'cryik , 

we  have  (since  the  functions  j  are  linearly  independent),  on  equating 
the  coefficients  of ///,«/,  the  system  of  relations 


506  LINEAR  DIFFERENTIAL  EQUATIONS. 

the  summations  extending  over  all  values  of  /  which  are  less  than  i 
and  greater  than  /',  and  over  the  corresponding  values  of  in. 
We  will  now  proceed  to  form  functions 

Jii ,  •  •  •  ,  jJ'.-i ,  .  .  . 

which  submit  to  the  substitutions  (34)  and  (35),  satisfying  the  con- 
dition (37)  or,  what  is  the  same  thing,  the  system  of  conditions  (41). 
Denoting  hy  a  2.  constant,  and  recalling  the  definition  given  above 
of  the  functions  0,  we  have 


(42) 
Write  now 

(43) 


Q(x  —  a  -\-  00)   =5,  0(-r  —  a), 

e{x  —  a-\-  co')  =  s/G{x  —  a). 


{>' 


I,  m 


In  these  equations  the  functions  Uil'  are  determinate  polynomials  of 
degree  /  —  /  in  /a  and  pi' ;  the  functions  0  are  ordinary  doubly- 
periodic  functions  with  periods  go  and  go',  i.e., 

0{x  -\-  go)  =  0{x),     0{x  -\-  go')  =  0{x) ; 

and  the  summations  extend  over  all  values  of  /  less  than  i  and  over 
the  corresponding  values  of  m.  We  have  now  to  show  that  the 
functions  J/  defined  by  these  equations  satisfy  all  the  prescribed  con- 
ditions and  are  integrals  of  the  given  differential  equation 

d^'y  d'^'^y 

where  the  coefHcients  /  are  uniform  and  doubly  periodic,  having  00 
and  &)'  for  periods.  From  equations  (42)  we  see  that  all  of  the 
functions  y  in  (43)  whose  first  suffix  is  unity  satisfy  the  conditions 
of  (34)  and  (35).     To  investigate  the  remaining  ones  we  will  write 

(44)  yik  =  0{x  —  a)Sik . 


DOUBLY-PERIODIC   COEFFICIENTS.  SO/" 

Now  since  the  change  of  x  into  x  -\-  g)  multiplies  Q{x  —  a)  by  s^ ,. 
and  the  change  of  x  into  x  -{-  go'  multiplies  this  same  function  by  s/, 
it  is  clear,  from  (34)  and  (35),  that  these  changes  must  cause  the 
functions  2  to  submit  to  the  substitutions 

(45)  ■^  ^  I  'S'lij    •  •  •  )   ^ik  f   •  •  •  ;  ^iA>    •  •  •  >  ^ik 'T' -^ik  1   •  •  •  I  V 

(46)  t'  =  I  ^i4,   .  .  .  ,   Sik,   .  •  •  ;  ^ik,   .  •  •  ,  2,/,-\-Z'i^,  .  .  .  I  , 

where  the  functions  Z^  ,  Z'^^  are  deduced  from  the  functions  F,^ ,  F',^ 
of  (34)  and  (35)  by  replacing  in  the  latter  the  functions  j/  by  the 
new  functions  2.    We  have  of  course,  from  (45)  and  (46),  the  relation 

(47)  tt'  =  r'r. 

As  all  the  functions  z  whose  first  suffix  is  unity  remain  unaltered 
by  the  substitutions  r  and  t',  they  are  ordinary  doubly-periodic 
functions,  and  we  can  therefore  write 

(48)  ^ik  =  ^lA ; 

and  consequently,  so  far  as  the  functions_yi4are  concerned,  all  the  re- 
quired conditions  are  satisfied.  Let  us  assume  that  we  have  succeeded 
step  by  step  in  constructing  all  of  the  functions  s  whose  first  sufifix 
is  less  than  A,  and  that  the  general  form  of  these  functions  is  given  by 

(49)  ^ik  =  ^i, + 2  mi:  ^,„,   {i  <  A). 

a,  a' 

We  have  now  to  construct  the  functions  z^.  such  that 

(50)  ZKk{x  +  &?)  =  z^k  +  Z^k ,     ZKk{x  +  go')  =  z^^.  +  Z\i . 

The  functions  Z  and  Z'  being  linear  functions  of  the  ^-'s  whose  first 
suffix  is  less  than  A,  we  have,  by  hypothesis, 

(51)  ^A>.  =  ^^110.„,    Z\,  =  ^H'1'2^,„., 

/,  '«  I,  m 

where  BkI  ^"d  ^  ^^  are  polynomials  of  order  A  —  i  —  I  In  /x  and  //' 
and  depend  linearly  on  the  coefficients  of  Z^k  and  Z\i,  and  where 
the  summations  extend  to  all  systems  of  values  of  /  and  7U  for  which 
/<  A. 


508  LINEAR   DIFFERENTIAL   EQUATIONS. 

The  substitutions  rr'  and  t't  applied  to  Zi^  give 

d' Zxk  denoting  the  increment  received  by  Z^^,  from  the  substitution 
t',  and  SZ'xk  denoting  the  increment  received  by  Z\k  from  the 
substitution  r.     Now  since  rr'  =  r'r,  we  must  have 

{53)  8'Z^  =  6Z'^. 

We  have  to  bear  in  mind  now  that  we  have  to  form  functions 
in  general  which  are  such  that  the  results  of  the  substitutions  r  and 
r'  shall  respectively  be  equal  to  the  results  arising  from  changing  x 
into  X  -\-  00  and  x  -\-  oo' .  Now  by  hypothesis  these  equalities  hold 
for  the  Zijlj  <  A)  already  constructed,  and  consequently  they  hold 
for  Zxk,  Z'xk,  which  are  linear  functions  of  the  same.  It  follows 
then  that 


(54) 


ly  ni 


(Since  ^  is  an  ordinary  doubly-periodic  function,  we  have  A0  =  A'^ 
=  o.)  From  (53)  we  must  now  have,  since  the  functions  ^;,„  are 
arbitrary, 

(55)  A'aki   =   A^    A^. 

The  conditions  that  these  equations  may  be  satisfied  are  given  in 
equations  (27),  and  in  equations  (28),  (29),  and  (30)  we  have  seen  that 
a  polynomial  TI'^  can  be  constructed  such  that 

(  /ITT'"'  —  '^'"' 

(56)  \  , ;  „ ; 

(z/'i7ll=H'll- 
Supposing  the  polynomials  77  to  be  so  constructed,  write 

<57)  z^k^U^k-\-^ntk^i>n\ 

I,  tn 


DOUBLY-PERIODIC  COEFFICIENTS.  509 

in  this  change  x  into  x  -{-  oj,  and  the  resulting  increment  is 

(58)  AU^-\-^  ^^'Ll^i'u  =^AU^-^^  Hl'^^/.«  =  ^t/^  +  Z^k- 

Again   changing  x  into  x  -\-  go,  we  have  as  the  corresponding  in- 
crement 

(59)  A'U^i-{-Z\i. 

From  equations  (50)  to  (55),  inclusive,  we  see  that  s^^  will  satisfy  all 
the  required  conditions  if  we  have 

(60)  A  [7x^  =  0     and     z/'t^A^  =  o; 

that  is,  if  [/\k  is  a  doubly-periodic  function  with  periods  go  and  co'.  We 
have  thus  established  the  fact  that  the  general  form  of  the  integrals 
jij(.  is  that  given  by  equations  (43).  In  order  to  satisfy  the  given 
differential  equation,  however,  we  must  still  determine  the  doubly- 
periodic  functions  0,4  and  find  the  values  of  all  the  constants  which 
have  entered  into  the  preceding  analysis.  We  know  in  the  first 
place  that  the  critical  points  of  the  integrals  are  those  of  the  coeffi- 
cients/,,  .  .  .  ,  p„  oi  the  differential  equation,  and  therefore,  since 
these  points  are  known,  we  know  all  the  critical  points  (in  our  case 
these  are  poles)  of  the  integrals.  Suppose  a,  d,  .  .  .to  denote  the 
poles  of  the  integrals.  By  the  test  which  we  suppose  to  have  been 
made  in  order  to  determine  whether  or  not  the  general  integral  of 
the  equation  is  uniform  (as  in  our  case  it  is  supposed  to  be)  and 
which  consists  in  developing  the  integral  according  to  increasing 
powers  of  x  —  a,  x  —  d,  .  .  .  ,  we  have  also  discovered  the  various 
orders  of  multiplicity,  say  a,  ^,  .  .  .  ,  of  these  poles.  We  know 
then  that  the  orders  of  multiplicity  of  the  poles  a,  d,  .  .  .in  the  par- 
ticular integrals  cannot  respectively  exceed  a,  fi,  .  .  .  Consider  now 
the  function  j,-^ ,  that  is, 

(61)  J,,  =  &{x  -  ^)[^,,  -f  ^  77,^-0^'"]  ■=  Q{x  -  a)Zi, . 

The  function  Q{x  —  a)  already  defined    admits   only  a   simple 
pole  rt*  and  a  simple  zero  a  -\-  q  \  the  constant  a,  which  has  so  far 

*  Q(x  —  a)  has  of  course,  in  addition  to  a,  the  poles  a  -{-m(o-\-  ni  go',  where  m  and 
m   are  integers;  but  we  confine  ourselves  to  the  fundamental  or  primary  pole  a. 


5IO  LINEAR  DIFFERENTIAL   EQUATIONS. 

TDeen  arbitrary,  will  now  be  taken  as  one  of  the  poles  of  the  coeffi- 
cients/. The  functions  Zi^  as  defined  by  (6i)  can  then  only  have  as 
poles  the  points  a-\-  q,  a,  b,  .  .  .  ,  with  orders  of  multiplicity  re- 
spectively equal  at  most  to  i,  or  —  i,  /?,  .  .  .  ;  further,  the  functions 
jx  and  yu'  have  only  the  simple  pole  of  the  function  Z{x  —  a),  viz., 
jx:  =:  a,  and  therefore  the  polynomials  77'4"  have  this  point  as  their 
only  pole,  and  its  order  of  multiplicity  is  i  —  I.     From  the  relations 

'(62)  2,,  =  ^,,  +  2  IV;S0„, 


we  now  see  at  once  that  the  functions  ^,-4  can  have  no  other  poles 
than  the  points 

a-^-q,     a,     b,     .  .  .  , 

and  that  these  have  respectively  orders  of  multiplicity  at  most 
equal  to 

I,     a  -\-i  ~  2,     fi,     .  .  . 

The  forms  of  these  functions  are  given  by  equations  (10)  and  (14). 
The  forms  of  the  integrals  j,-^  are  now  known,  but  we  have  still  to 
determine  the  parameters  p  and  q  and  the  constant  coefficients 
which  enter  into  the  expressions  of  the  functions  ^,4  and  the  linear 
functions  F^-^ .  The  necessity  for  determining  the  constants  in  the 
functions  F,-^  arises  from  the  fact  that  the  polynomials  77  depend 
upon  these  functions.  These  constants  will  be  determined  by  sub- 
stituting in  the  differential  equation  the  expressions  above  found  for 
the  integrals  jj/,-^ . 

Consider  first  the  integrals  which  are  doubly-periodic  functions 
of  the  second  kind,  i.e.,  the  integrals 

7n ,    J12  •  .  •  yik  .  .  .  ,     (multipliers  s^  and  s,',) 
z,, ,     z,^  .  .  .  z,k  .  .  .  ,     (multipliers  s,  and  s^\) 

Take  first  the  integral  jj/,,  given  by  the  equation 
(63)  :y,,  =  &{x-a)0,,  =  0^, 


A 


DOUBL  Y-FER IODIC  COEFFICIENTS. 
"where  for  brevity  we  have  written 

Bjx  -a-q) 


511 


(64)       ^  =  ^n     and     0  =  Q{x  —  a),  =  ^^<*- 

The  logarithmic  derivative  of  Q  is 
0' 


d,{x  —  a) 


(65) 


0 


=  /  +  Z{x  —  a  —  q)  —  Z{x  —  «), 


or,  from  equation  (12), 


sn  q 


^     ^        0       ^        sn(^  —  a)sn{x  —  a — q)  \2 


^{^-9]-z 


Since  00  and  00'  are  the  periods  of  the  eUiptic  function  sn  /,  the 
second  member  of  this  equation  is  a  doubly-periodic  function  which 
we  will  denote  by  D.,  that  is, 


(67)       /  +  - 


sn  q 


sn  {x  —  d)'i\\{x—a  —  q) 


+  z(^-,l-z(^)=a 


(68) 


©' 


From  this  last  equation  we  have 

(69)  &  =  0/2, 

and  consequently 

d 


(70) 


^^&^  =  0'^  +  0^'  =  e\n^-j-0'\, 

^0^  =  0jn[O^+0']  +  [n^+^']'f 

=  0][/2^  +  i2']^  + 2/2^' +  (?"}, 


Substituting  these  values  in  the  first  member  of  the  differential 
equation 

d"y  d"-y 


512 


LINEAR   DIFFERENTIAL   EQUATIONS. 


we  have  as  the  result  the  expression  QA,  denoting  by  A  the  doubly- 
periodic  function 

(71)    J=A^+A-il^^+^'} 

The  coefficients/;  are  doubly-periodic  functions  which  by  hypothesis 
have  the  points  a,b,...2A  poles  of  orders  of  multiplicity  at  most 
=  /,  and  so  from  (14)  they  are  of  the  form 


(72)    Pi^Ai---^ 


sn<7; 


sn;rsn(;ir — a) 
sn  b 


-A. 


sn:rsn(;r  — <^) 


B„ 


'~,n\x—a) 
I 


A 


d' 


%Vl^{x — a) 


dx'-  ^sv^ix—d) 


In  equation  (14)  we  have  a  similar  form  for  <?  which  is  linear  and 
homogeneous  in  the  as  yet  unknown  constant  coefficients 


A^,  A^, 


,     B,,  B,,  .  .  .,  M,  C. 


From  (14)  we  deduce  by  differentiation  the  values  of  <?',   <?" , 
Finally,  if  we  write,  for  brevity, 


(73) 
we  have 

(74) 


/=^  +  ^(t-^)-^(t> 


n=p'  + 


snq 


sn{x — a)  sn  (;f  —  a  —  q)^ 
\  n',=  Z'{x  -  a  —  q)-  Z'{x  -  d), 
I 


+ 


I 


sn'(-r  —  a  —  q)        s\\'{x  —  d) 

and  from  this  last  equation  we  derive  at  once  the  values  of 

£1",      a"\      .... 

Substituting  these  values  in  (71),  we  obtain  the  final  expression  for 
the  doubly-periodic  function  A.  If  now  j,,  is  an  integral  of  the 
equation,  A  must  vanish,  and  by  a  known  theorem  A  will  vanish  if 
we  can  show  that  it  has  more  zeros  than  it  has  poles. 


DOUBLY-PERIODIC  COEFFICIENTS.  513 

The  function  j„  has  the  points  a,  b,  .  .  .  ■SiS,  poles  with  orders  of 

d^  y 
multipHcity  at  most  equal  \.o  a,  ^,  .  .  . ,  and  consequently  -~:  has 

dx' 

these  same  points  as  poles  of  orders  of  multiplicity  at  most  equal  to 

d'^y 
a  ^  i-  §  -\-  h  •  •  •  '      If  now  we  multiply  -—4-  by  the  coefficient 

dx^ 

p„_i,  we  obtain  as  the  product  an  expression  which  has  the  points 
a,  b,  .  .  .  2iS  poles  of  orders  of  multiplicity  at  most  equal  to 

a  -\-  n,      (5  +  71,      .... 

It  follows  then  that  SA,  which  is  a  sum  of  such  expressions,  pos- 
sesses the  same  property.  Further,  0  never  has  more  than  one  pole 
and  one  zero,  and  so  the  total  number  of  poles  of  A,  when  we  take 
account  of  their  orders  of  multiplicity,  is  at  most  equal  to 

a-\-n-{-/3-\-7i-\-.  .  .  . 

In  order  then  that  A  may  vanish  identically  it  is  only  necessary  to 
show  that  it  admits  of  arbitrarily  chosen  zeros  the  sum  of  whose 
orders  of  multiplicity  is  greater  than 

a  +  n  -\-  ^  +  71  -{-  .  .  .  ; 

or,  in  other  words,  if  the  sum  of  the  orders  of  multipHcity  of  the 
poles  of  A  is  less  than  d,  there  must  exist 

a-\-n-\-f3-\-n-{-  .  .  .   — d-j-i  zeros. 

The  system  of  equations  so  obtained  is  generally  superabundant, 
but  we  nevertheless  know  a  priori  that  they  admit  of  solutions. 
These  equations  are  linear  and  homogeneous  with  respect  to  the 
constants 

A,,  A,,  .  .  .  ,     B,,  B,,  .  .  .  ,    M,     C 

of  (P.  If  now  we  eliminate  these  constants,  which  may  be  done  in 
different  ways,  we  obtain  algebraic  relations  connecting/',  sn  ^  and 
its  derivative  cn^dn^.  To  each  such  system  of  values  of  /'  and  q 
corresponds  for/  a  value 

(75)  /=/  +  z(^)-z(^-,). 


514  LINEAR  DIFFERENTIAL  EQUATIONS. 

and  for  s^  and  s^  the  values 

(76)  s,  =  ^^",      5/=  ^^"  +    <-   . 

Substituting  the  values  of  p'  and  q  in  the  equations  of  condition, 
these  will  determine  certain  of  the  coef^cients 

A,,   A,,   .  .  .,     B,,   B,,  .  .  .,      M,      C 

in  terms  of  the  remaining  ones,  and  if  these  remaining  ones  are  v  in 
number,  we  shall  have  v  particular  integrals  j/„  ,  J12 ,  •  •  •  ,  yxv,  which 
are  doubly-periodic  functions  of  the  second  kind  admitting  5,  and  s^ 
as  multipliers.  We  will  thus  have  obtained  all  of  the  integrals 
which  are  doubly-periodic  functions  of  the  second  kind.  To  another 
system  of  values  of  p'  and  q  correspond  other  values  of  the  mul- 
tipliers, say  s^  and  5/,  etc.  We  determine  then  in  the  above  manner 
all  the  pairs 

?       ?'•        V       ?'*        c      ?'■  .. 

of  multipliers  and  the  corresponding  integrals 

y-i.\.  ■>  •  •  •  J     y\k )  •  •  •  > 

«  ^W    >       •      •      •    7  ^ik  >     •      •      •    > 


which  are  doubly  periodic  of  the  second  kind.  If  the  number  of 
these  integrals  is  equal  to  n  (which  is  generally  the  case),  their  hnear 
combination  will  give  the  general  integral ;  if  this  number  is  less 
than  n,  we  have  still  to  find  some  integrals.  Let  us  suppose  that  we 
have  constructed  all  of  the  integrals  jj/,a  ,  Zi^ ,  .  .  .  whose  first  index 
is  less  than  A,  and  that  we  have  determined  the  corresponding  linear 
functions  F.-^ ,  F,/,  ....  We  seek  now  to  determine  the  integrals 
yxk  (if  such  exist)  and  the  corresponding  functions  Fa^,  Fa/,  .... 
Now  we  know  that 

{77)  yu  =  0[^A.  -^2n'-<P,„,i 

where,  remembering  the  limits  of  the  summation,  everything  is 
known  save  the  indeterminate  coefificients 


DOUBLY-PERIODIC  COEFFICIENTS.  515 

of  ^xk  and  the  coefficients  of  F^.^,  Y\k  which  enter  linearly  into 
the  polynomials  77^'" . 

Substituting  the  above  expression  for  y,^k  in  the  given  differen- 
tial equation,  we  obtain  as  the  result  OA^^,  where  A\i  is  a  function 
such  that  the  sum  of  the  orders  of  multiplicity  of  its  poles  is  not 
greater  than 

a-\-7i-\-/3-\-n-\-  .  .  .  . 

Further,  Ak^  is  doubly  periodic :  to  show  this  it  is  only  necessary  to 
change  x  into  x  -\-  go;  the  result  of  this  change  is  obviously  the  same 
whether  it  be  made  in  y^k  before  the  substitution  in  the  differential 
equation  or  be  made  in  &A\k  after  the  substitution.  The  result  of 
changing  x  into  x  -{-  go  \s  to  change  Ja^  into  s,{y^^  -\-  Y^^^) ;  and 
since  Y^k  is  a  linear  function  of  the  integrals  y  already  found,  it  fol- 
lows that  on  substituting  s^{yKk  +  YKk)  in  the  differential  equation 
the  result  will  be 

S,&AKk- 

But  Q{x -\-  00)  =  s^O{x),  and  consequently 

AKi{x  -{-&?)  =  A!,i{x). 

A  similar  result  will  obviously  be  obtained  if  we  change  x  into 
X  -f-  00';  that  is,  we  should  find 

and  therefore  A\j^  is  a  doubly-periodic  function.  In  order  now  that 
Ayi  may  vanish  (as  it  must  if  j^^  is  an  integral)  it  is  only  necessary 
to  show,  in  the  manner  already  described,  that  the  sum  of  the  orders 
of  multiplicity  of  its  zeros  is  greater  than 

a  ^  n  -\-  /3  -\-  n  -\-  .  .  .  . 

We  will  thus  determine  a  system  of  linear  homogeneous  equations 
for  the  determination  of  the  unknown  coefficients.  If  this  system 
is  compatible,  there  will  still  exist,  and  we  can  determine,  new  inte- 
grals y^k ;  if  it  be  not  compatible,  we  will  know  that  the  integrals 
J';/,  of  the  first  class  have  been  all  determined,  and  in  order  to  obtain 
new  integrals  we  shall  have  to  start  with  those  of  the  second  class. 


5l6  '  LINEAR  DIFFERENTIAL  EQUATIONS. 

viz.,  the  integrals  Zik,  etc.     This  process  being  continued,  we  will 
finally  arrive  at  a  system  of  7t  linearly  independent  integrals. 
Lamp's  equation 

-~  —  \jn{in  —  i)/^'sn';r  -\-  /i\y  =  o 

{m  a  positive  integer,  h  an  arbitrary  constant,  and  k  the  modulus  of 
the  elliptic  function  sn  x)  is  probably  the  most  important  known 
equation  of  the  type  above  considered.  It  is  not  possible,  however, 
in  the  limits  of  this  treatise  to  investigate  this  equation,  and  so  the 
reader  is  referred  to  the  various  papers  on  the  subject  by  M.  Her- 
mite  in  the  Comptes  Rendus  of  the  Academy  of  Sciences  (Paris) 
in  Crelle's  Journal,  and  particularly  to  M.  Hermite's  treatise  "■  Sur 
quelques  applications  dcs  Fonctions  Elliptiques'  (Paris  :  Gauthier-Villars, 
1885).  Two  memoirs  by  Count  de  Sparre  in  Vol.  Ill  of  the  Acta 
Mathematica  should  also  be  consulted.  The  common  title  of  de 
Sparre's  two  memoirs  is 
"  Siir  r Equation 


dy 

dx'^ 


k'  sn  X  en  X  sn  ;r  dn  ;ir  en  ;r  dn  x~\  dy 

2v \ —4-  2y, 2v. 


dn  ;r  *      en  ;r  ^      en  x 


dx 


A     ■* 

{ri,  -  rX^i,  +  r,  +  I)  +  —r-  (n,  -  i/,)(;/,  +  r.  +  l) 


Lsn'  X  ^  '         -/x  -    .  '    .    en*  x 

+  ^'  d':^  («.  -  ^)(«>  +  y+i)  +  ^'  sn^  x{n  +  ,.  +  r,  +  r,) 

(«  —  r  —  Vj  —  r,  +  i)  -f  /l']j'. 

Equation  oil  v.,  r, ,  v^  d^signent  des  nombres  quelconques,  n,  n^ ,  ;/, ,  n^  dcs 
noinbres  entiers  positive  ou  negative,  et  h  une  constante  arbitrairey 

The  subject  of  equations  with  doubly-periodic  coefificients  will 
be  resumed  later  in  connection  with  the  invariantive  theory ;  the 
reader  is  here,  however,  advised  to  consult  Halphen's  "  Mdmoire  sur  la 
reduction  des  Equations  differ  entie  lies  lindaires  aux  Formes  integrables'' 
(Savants  Etrangferes,  vol.  xxviii),  and  also  Chapter  XIII  of  the  second 
volume  of  his  "  Traits  des  Fo)ictions  Elliptiqucs." 

END   OF  VOL.   I. 


V\^'''^'' 


t 


TTNTV 


^Y  OF  f      TT-oR- 


TBRARY 


*LcaiA»aiM*aji6fifc:ii^--l.^-;';l?S'. 


14  DAY  USE 

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